Probability Questions with Solutions

Tutorial on finding the probability of an event. In what follows, S is the sample space of the experiment in question and E is the event of interest. n(S) is the number of elements in the sample space S and n(E) is the number of elements in the event E.

Questions and their Solutions

Answers to the above exercises, more references and links.

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Probability Practice Problems

1. on a six-sided die, each side has a number between 1 and 6. what is the probability of throwing a 3 or a 4, 2. three coins are tossed up in the air, one at a time. what is the probability that two of them will land heads up and one will land tails up, 3. a two-digit number is chosen at random. what is the probability that the chosen number is a multiple of 7, 4. a bag contains 14 blue, 6 red, 12 green, and 8 purple buttons. 25 buttons are removed from the bag randomly. how many of the removed buttons were red if the chance of drawing a red button from the bag is now 1/3, 5. there are 6 blue marbles, 3 red marbles, and 5 yellow marbles in a bag. what is the probability of selecting a blue or red marble on the first draw, 6. using a six-sided die, carlin has rolled a six on each of 4 successive tosses. what is the probability of carlin rolling a six on the next toss, 7. a regular deck of cards has 52 cards. assuming that you do not replace the card you had drawn before the next draw, what is the probability of drawing three aces in a row.

  • 1 in 132600

8. An MP3 player is set to play songs at random from the fifteen songs it contains in memory. Any song can be played at any time, even if it is repeated. There are 5 songs by Band A, 3 songs by Band B, 2 by Band C, and 5 by Band D. If the player has just played two songs in a row by Band D, what is the probability that the next song will also be by Band D?

  • Not enough data to determine.

9. Referring again to the MP3 player described in Question 8, what is the probability that the next two songs will both be by Band B?

10. if a bag of balloons consists of 47 white balloons, 5 yellow balloons, and 10 black balloons, what is the approximate likelihood that a balloon chosen randomly from the bag will be black, 11. in a lottery game, there are 2 winners for every 100 tickets sold on average. if a man buys 10 tickets, what is the probability that he is a winner, answers and explanations.

1.  B:  On a six-sided die, the probability of throwing any number is 1 in 6. The probability of throwing a 3 or a 4 is double that, or 2 in 6. This can be simplified by dividing both 2 and 6 by 2.

Therefore, the probability of throwing either a 3 or 4 is 1 in 3.

2.  D:  Shown below is the sample space of possible outcomes for tossing three coins, one at a time. Since there is a possibility of two outcomes (heads or tails) for each coin, there is a total of 2*2*2=8 possible outcomes for the three coins altogether. Note that H represents heads and T represents tails:

HHH HHT HTT HTH TTT TTH THT THH

Notice that out of the 8 possible outcomes, only 3 of them (HHT, HTH, and THH) meet the desired condition that two coins land heads up and one coin lands tails up. Probability, by definition, is the number of desired outcomes divided by the number of possible outcomes. Therefore, the probability of two heads and one tail is 3/8, Choice D.

3.  E:  There are 90 two-digit numbers (all integers from 10 to 99). Of those, there are 13 multiples of 7: 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98.

4.  B:  Add the 14 blue, 6 red, 12 green, and 8 purple buttons to get a total of 40 buttons. If 25 buttons are removed, there are 15 buttons remaining in the bag. If the chance of drawing a red button is now 1/3, then 5 of the 15 buttons remaining must be red. The original total of red buttons was 6. So, one red button was removed.

5.  D:  Use this ratio for probability:

Probability = Number of Desired Outcomes

Number of Possible Outcomes

There are 6 blue marbles and 3 red marbles for a total of 9 desired outcomes. Add the total number of marbles to get the total number of possible outcomes, 14. The probability that a red or blue marble will be selected is 9/14.

6.  C:  The outcomes of previous rolls do not affect the outcomes of future rolls. There is one desired outcome and six possible outcomes. The probability of rolling a six on the fifth roll is 1/6, the same as the probability of rolling a six on any given individual roll.

7.  D:  The probability of getting three aces in a row is the product of the probabilities for each draw. For the first ace, that is 4 in 52 or 1 in 13; for the second, it is 3 in 51 or 1 in 27; and for the third, it is 2 in 50 or 1 in 25. So the overall probability,  P , is P=1/13*1/17*1/25=1/5,525

8.  B:  The probability of playing a song by a particular band is proportional to the number of songs by that band divided by the total number of songs, or 5/15=1/3 for B and D. The probability of playing any particular song is not affected by what has been played previously, since the choice is random and songs may be repeated.

9.  A:  Since 3 of the 15 songs are by Band B, the probability that any one song will be by that band is 3/15=1/5. The probability that the next two songs are by Band B is equal to the product of two probabilities, where each probability is that the next song is by Band B: 1/5*1/5=1/25 The same probability of 1/5 may be multiplied twice because whether or not the first song is by Band B has no impact on whether the second song is by Band B. They are independent events.

10.  B:  First, calculate the total number of balloons in the bag: 47 + 5 + 10 = 62.

Ten of these are black, so divide this number by 62. Then, multiply by 100 to express the probability as a percentage:

10 / 62 = 0.16

0.16 100 = 16%

11. C: First, simplify the winning rate. If there are 2 winners for every 100 tickets, there is 1 winner for every 50 tickets sold. This can be expressed as a probability of 1/50 or 0.02. In order to account for the (unlikely) scenarios of more than a single winning ticket, calculate the probability that none of the tickets win and then subtract that from 1. There is a probability of 49/50 that a given ticket will not win. For all ten to lose that would be (49/50)^(10) ≈ 0.817. Therefore, the probability that at least one ticket wins is 1 − 0.817 = 0.183 or about 18.3%

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15 Probability Questions And Practice Problems for Middle and High School: Harder Exam Style Questions Included

Beki christian.

Probability questions and probability problems require students to work out how likely it is that something is to happen. Probabilities can be described using words or numbers. Probabilities range from 0 to 1 and can be written as fractions, decimals or percentages .

Here you’ll find a selection of probability questions of varying difficulty showing the variety you are likely to encounter in middle school and high school, including several harder exam style questions.

What are some real life examples of probability?

How to calculate probabilities, probability question: a worked example, 6th grade probability questions, 7th grade probability questions, 8th grade probability questions, 9th grade probability questions, 10th grade probability questions.

  • 11th & 12th grade probability questions

Looking for more middle school and high school probability math questions?

The more likely something is to happen, the higher its probability. We think about probabilities all the time.

For example, you may have seen that there is a 20% chance of rain on a certain day or thought about how likely you are to roll a 6 when playing a game, or to win in a raffle when you buy a ticket.

15 Probability Questions Worksheet

Want the 15 multiple choice probability questions from this blog in a handy downloadable format? Then look no further!

The probability of something happening is given by:

We can also use the following formula to help us calculate probabilities and solve problems:

  • Probability of something not occuring = 1 – probability of if occurring P(not\;A) = 1 - P(A)
  • For mutually exclusive events: Probability of event A OR event B occurring = Probability of event A + Probability of event B P(A\;or\;B) = P(A)+P(B)
  • For independent events: Probability of event A AND event B occurring = Probability of event A times probability of event B P(A\;and\;B) = P(A) × P(B)

Question: What is the probability of getting heads three times in a row when flipping a coin?

When flipping a coin, there are two possible outcomes – heads or tails. Each of these options has the same probability of occurring during each flip. The probability of either heads or tails on a single coin flip is ½.

Since there are only two possible outcomes and they have the same probability of occurring, this is called a binomial distribution.

Let’s look at the possible outcomes if we flipped a coin three times.

Let H=heads and T=tails.

The possible outcomes are: HHH, THH, THT, HTT, HHT, HTH, TTH, TTT

Each of these outcomes has a probability of ⅛.

Therefore, the probability of flipping a coin three times in a row and having it land on heads all three times is ⅛.

Middle school probability questions

In middle school, probability questions introduce the idea of the probability scale and the fact that probabilities sum to one. We look at theoretical and experimental probability as well as learning about sample space diagrams and venn diagrams.

1. Which number could be added to this spinner to make it more likely that the spinner will land on an odd number than a prime number?

GCSE Quiz False

Currently there are two odd numbers and two prime numbers so the chances of landing on an odd number or a prime number are the same. By adding 3, 5 or 11 you would be adding one prime number and one odd number so the chances would remain equal.

By adding 9 you would be adding an odd number but not a prime number. There would be three odd numbers and two prime numbers so the spinner would be more likely to land on an odd number than a prime number.

2. Ifan rolls a fair dice, with sides labeled A, B, C, D, E and F. What is the probability that the dice lands on a vowel?

A and E are vowels so there are 2 outcomes that are vowels out of 6 outcomes altogether.

Therefore the probability is   \frac{2}{6} which can be simplified to \frac{1}{3} .

3. Max tested a coin to see whether it was fair. The table shows the results of his coin toss experiment:

Heads          Tails

    26                  41

What is the relative frequency of the coin landing on heads?

Max tossed the coin 67 times and it landed on heads 26 times.

\text{Relative frequency (experimental probability) } = \frac{\text{number of successful trials}}{\text{total number of trials}} = \frac{26}{67}

4. Grace rolled two dice. She then did something with the two numbers shown. Here is a sample space diagram showing all the possible outcomes:

What did Grace do with the two numbers shown on the dice?

Add them together

Subtract the number on dice 2 from the number on dice 1

Multiply them

Subtract the smaller number from the bigger number

For each pair of numbers, Grace subtracted the smaller number from the bigger number.

For example, if she rolled a 2 and a 5, she did 5 − 2 = 3.

5. Alice has some red balls and some black balls in a bag. Altogether she has 25 balls. Alice picks one ball from the bag. The probability that Alice picks a red ball is x and the probability that Alice picks a black ball is 4x. Work out how many black balls are in the bag.

Since the probability of mutually exclusive events add to 1:  

\begin{aligned} x+4x&=1\\\\ 5x&=1\\\\ x&=\frac{1}{5} \end{aligned}

\frac{1}{5} of the balls are red and \frac{4}{5} of the balls are blue.

6. Arthur asked the students in his class whether they like math and whether they like science. He recorded his results in the venn diagram below.

How many students don’t like science?

We need to look at the numbers that are not in the ‘Like science’ circle. In this case it is 9 + 7 = 16.

High school probability questions

In high school, probability questions involve more problem solving to make predictions about the probability of an event. We also learn about probability tree diagrams, which can be used to represent multiple events, and conditional probability.

7. A restaurant offers the following options:

Starter – soup or salad

Main – chicken, fish or vegetarian

Dessert – ice cream or cake

How many possible different combinations of starter, main and dessert are there?

The number of different combinations is 2 × 3 × 2 = 12.

8. There are 18 girls and 12 boys in a class. \frac{2}{9} of the girls and \frac{1}{4} of the boys walk to school. One of the students who walks to school is chosen at random. Find the probability that the student is a boy. 

First we need to work out how many students walk to school:

\frac{2}{9} \text{ of } 18 = 4

\frac{1}{4} \text{ of } 12 = 3

7 students walk to school. 4 are girls and 3 are boys. So the probability the student is a boy is \frac{3}{7} .

9. Rachel flips a biased coin. The probability that she gets two heads is 0.16. What is the probability that she gets two tails?

We have been given the probability of getting two heads. We need to calculate the probability of getting a head on each flip.

Let’s call the probability of getting a head p.

The probability p, of getting a head AND getting another head is 0.16.

Therefore to find p:

The probability of getting a head is 0.4 so the probability of getting a tail is 0.6.

The probability of getting two tails is 0.6 × 0.6 = 0.36 .

10. I have a big tub of jelly beans. The probability of picking each different color of jelly bean is shown below:

If I were to pick 60 jelly beans from the tub, how many orange jelly beans would I expect to pick?

First we need to calculate the probability of picking an orange. Probabilities sum to 1 so 1 − (0.2 + 0.15 + 0.1 + 0.3) = 0.25.

The probability of picking an orange is 0.25.

The number of times I would expect to pick an orange jelly bean is 0.25 × 60 = 15 .

11. Dexter runs a game at a fair. To play the game, you must roll a dice and pick a card from a deck of cards.

To win the game you must roll an odd number and pick a picture card. The game can be represented by the tree diagram below.

Dexter charges players $1 to play and gives $3 to any winners. If 260 people play the game, how much profit would Dexter expect to make?

Completing the tree diagram:

Probability of winning is \frac{1}{2} \times \frac{4}{13} = \frac{4}{26}

If 260 play the game, Dexter would receive $260.

The expected number of winners would be \frac{4}{26} \times 260 = 40

Dexter would need to give away 40 × $3 = $120 .

Therefore Dexter’s profit would be $260 − $120 = $140.

12. A fair coin is tossed three times. Work out the probability of getting two heads and one tail.

There are three ways of getting two heads and one tail: HHT, HTH or THH.

The probability of each is \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} 

Therefore the total probability is \frac{1}{8} +\frac{1}{8} + \frac{1}{8} = \frac{3}{8}

11th/12th grade probability questions

13. 200 people were asked about which athletic event they thought was the most exciting to watch. The results are shown in the table below.

A person is chosen at random. Given that that person chose 100m, what is the probability that the person was female?

Since we know that the person chose 100m, we need to include the people in that column only.

In total 88 people chose 100m so the probability the person was female is \frac{32}{88}   .

14.   Sam asked 50 people whether they like vegetable pizza or pepperoni pizza.

37 people like vegetable pizza. 

25 people like both. 

3 people like neither.

Sam picked one of the 50 people at random. Given that the person he chose likes pepperoni pizza, find the probability that they don’t like vegetable pizza.

We need to draw a venn diagram to work this out.

We start by putting the 25 who like both in the middle section. The 37 people who like vegetable pizza includes the 25 who like both, so 12 more people must like vegetable pizza. 3 don’t like either. We have 50 – 12 – 25 – 3 = 10 people left so this is the number that must like only pepperoni.

There are 35 people altogether who like pepperoni pizza. Of these, 10 do not like vegetable pizza. The probability is   \frac{10}{35} .

15. There are 12 marbles in a bag. There are n red marbles and the rest are blue marbles. Nico takes 2 marbles from the bag. Write an expression involving n for the probability that Nico takes one red marble and one blue marble.

We need to think about this using a tree diagram. If there are 12 marbles altogether and n are red then 12-n are blue.

To get one red and one blue, Nico could choose red then blue or blue then red so the probability is:

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The content in this article was originally written by secondary teacher Beki Christian and has since been revised and adapted for US schools by elementary math teacher Katie Keeton.

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  • Math Article

Probability

Probability  means possibility. It is a branch of mathematics that deals with the occurrence of a random event. The value is expressed from zero to one. Probability has been introduced in Maths to predict how likely events are to happen. The meaning of probability is basically the extent to which something is likely to happen. This is the basic probability theory, which is also used in the  probability distribution , where you will learn the possibility of outcomes for a random experiment. To find the probability of a single event to occur, first, we should know the total number of possible outcomes.

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Probability Definition in Math

Probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total certainty. We can predict only the chance of an event to occur i.e., how likely they are going to happen, using it. Probability can range from 0 to 1, where 0 means the event to be an impossible one and 1 indicates a certain event. Probability for Class 10 is an important topic for the students which explains all the basic concepts of this topic. The probability of all the events in a sample space adds up to 1.

For example , when we toss a coin, either we get Head OR Tail, only two possible outcomes are possible (H, T). But when two coins are tossed then there will be four possible outcomes,  i.e {(H, H), (H, T), (T, H),  (T, T)}.

Download this lesson as PDF: – Download PDF Here

Formula for Probability

The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favourable outcomes and the total number of outcomes.

Sometimes students get mistaken for “favourable outcome” with “desirable outcome”. This is the basic formula. But there are some more formulas for different situations or events.

Solved Examples

1) There are 6 pillows in a bed, 3 are red, 2 are yellow and 1 is blue. What is the probability of picking a yellow pillow?

Ans: The probability is equal to the number of yellow pillows in the bed divided by the total number of pillows, i.e. 2/6 = 1/3.

2) There is a container full of coloured bottles, red, blue, green and orange. Some of the bottles are picked out and displaced. Sumit did this 1000 times and got the following results:

  • No. of blue bottles picked out: 300
  • No. of red bottles: 200
  • No. of green bottles: 450
  • No. of orange bottles: 50

a) What is the probability that Sumit will pick a green bottle?

Ans: For every 1000 bottles picked out, 450 are green.

Therefore, P(green) = 450/1000 = 0.45

b) If there are 100 bottles in the container, how many of them are likely to be green?

Ans: The experiment implies that 450 out of 1000 bottles are green.

Therefore, out of 100 bottles, 45 are green.

Probability Tree

The tree diagram helps to organize and visualize the different possible outcomes. Branches and ends of the tree are two main positions. Probability of each branch is written on the branch, whereas the ends are containing the final outcome. Tree diagrams are used to figure out when to multiply and when to add. You can see below a tree diagram for the coin:

Probability Tree

Types of Probability

There are three major types of probabilities:

Theoretical Probability

Experimental probability, axiomatic probability.

It is based on the possible chances of something to happen. The theoretical probability is mainly based on the reasoning behind probability. For example, if a coin is tossed, the theoretical probability of getting a head will be ½.

It is based on the basis of the observations of an experiment. The experimental probability can be calculated based on the number of possible outcomes by the total number of trials. For example, if a coin is tossed 10 times and head is recorded 6 times then, the experimental probability for heads is 6/10 or, 3/5.

In axiomatic probability, a set of rules or axioms are set which applies to all types. These axioms are set by Kolmogorov and are known as Kolmogorov’s three axioms. With the axiomatic approach to probability, the chances of occurrence or non-occurrence of the events can be quantified. The axiomatic probability lesson covers this concept in detail with Kolmogorov’s three rules (axioms) along with various examples.

Conditional Probability is the likelihood of an event or outcome occurring based on the occurrence of a previous event or outcome.

Probability of an Event

Assume an event E can occur in r ways out of a sum of n probable or possible equally likely ways . Then the probability of happening of the event or its success is expressed as;

The probability that the event will not occur or known as its failure is expressed as:

P(E’) = (n-r)/n = 1-(r/n)

E’ represents that the event will not occur.

Therefore, now we can say;

P(E) + P(E’) = 1

This means that the total of all the probabilities in any random test or experiment is equal to 1.

What are Equally Likely Events?

When the events have the same theoretical probability of happening, then they are called equally likely events. The results of a sample space are called equally likely if all of them have the same probability of occurring. For example, if you throw a die, then the probability of getting 1 is 1/6. Similarly, the probability of getting all the numbers from 2,3,4,5 and 6, one at a time is 1/6. Hence, the following are some examples of equally likely events when throwing a die:

  • Getting 3 and 5 on throwing a die
  • Getting an even number and an odd number on a die
  • Getting 1, 2 or 3 on rolling a die

are equally likely events, since the probabilities of each event are equal.

Complementary Events

The possibility that there will be only two outcomes which states that an event will occur or not. Like a person will come or not come to your house, getting a job or not getting a job, etc. are examples of complementary events. Basically, the complement of an event occurring in the exact opposite that the probability of it is not occurring. Some more examples are:

  • It will rain or not rain today
  • The student will pass the exam or not pass.
  • You win the lottery or you don’t.

Also, read: 

  • Independent Events
  • Mutually Exclusive Events

Probability Theory

Probability theory had its root in the 16th century when J. Cardan, an Italian mathematician and physician, addressed the first work on the topic, The Book on Games of Chance. After its inception, the knowledge of probability has brought to the attention of great mathematicians. Thus, Probability theory is the branch of mathematics that deals with the possibility of the happening of events. Although there are many distinct probability interpretations, probability theory interprets the concept precisely by expressing it through a set of axioms or hypotheses. These hypotheses help form the probability in terms of a possibility space, which allows a measure holding values between 0 and 1. This is known as the probability measure, to a set of possible outcomes of the sample space.

Probability Density Function

The Probability Density Function (PDF) is the probability function which is represented for the density of a continuous random variable lying between a certain range of values. Probability Density Function explains the normal distribution and how mean and deviation exists. The standard normal distribution is used to create a database or statistics, which are often used in science to represent the real-valued variables, whose distribution is not known.

Probability Terms and Definition

Some of the important probability terms are discussed here:

Probability of an Event

Applications of Probability

Probability has a wide variety of applications in real life. Some of the common applications which we see in our everyday life while checking the results of the following events:

  • Choosing a card from the deck of cards
  • Flipping a coin
  • Throwing a dice in the air
  • Pulling a red ball out of a bucket of red and white balls
  • Winning a lucky draw

Other Major Applications of Probability

  • It is used for risk assessment and modelling in various industries
  • Weather forecasting or prediction of weather changes
  • Probability of a team winning in a sport based on players and strength of team
  • In the share market, chances of getting the hike of share prices

Problems and Solutions on Probability

Question 1: Find the probability of ‘getting 3 on rolling a die’.

Sample Space = S = {1, 2, 3, 4, 5, 6}

Total number of outcomes = n(S) = 6

Let A be the event of getting 3.

Number of favourable outcomes = n(A) = 1

i.e. A  = {3}

Probability, P(A) = n(A)/n(S) = 1/6

Hence, P(getting 3 on rolling a die) = 1/6

Question 2: Draw a random card from a pack of cards. What is the probability that the card drawn is a face card?

A standard deck has 52 cards.

Total number of outcomes = n(S) = 52

Let E be the event of drawing a face card.

Number of favourable events = n(E) = 4 x 3 = 12 (considered Jack, Queen and King only)

Probability, P = Number of Favourable Outcomes/Total Number of Outcomes

P(E) = n(E)/n(S)

P(the card drawn is a face card) = 3/13

Question 3: A vessel contains 4 blue balls, 5 red balls and 11 white balls. If three balls are drawn from the vessel at random, what is the probability that the first ball is red, the second ball is blue, and the third ball is white?

The probability to get the first ball is red or the first event is 5/20.

Since we have drawn a ball for the first event to occur, then the number of possibilities left for the second event to occur is 20 – 1 = 19.

Hence, the probability of getting the second ball as blue or the second event is 4/19.

Again with the first and second event occurring, the number of possibilities left for the third event to occur is 19 – 1 = 18.

And the probability of the third ball is white or the third event is 11/18.

Therefore, the probability is 5/20 x 4/19 x 11/18 = 44/1368 = 0.032.

Or we can express it as: P = 3.2%.

Question 4: Two dice are rolled, find the probability that the sum is:

  • less than 13

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Probability Problems

  • Two dice are thrown together. Find the probability that the product of the numbers on the top of the dice is: (i) 6 (ii) 12 (iii) 7
  • A bag contains 10 red, 5 blue and 7 green balls. A ball is drawn at random. Find the probability of this ball being a (i) red ball (ii) green ball (iii) not a blue ball
  • All the jacks, queens and kings are removed from a deck of 52 playing cards. The remaining cards are well shuffled and then one card is drawn at random. Giving ace a value 1 similar value for other cards, find the probability that the card has a value (i) 7 (ii) greater than 7 (iii) less than 7
  • A die has its six faces marked 0, 1, 1, 1, 6, 6. Two such dice are thrown together and the total score is recorded. (i) How many different scores are possible? (ii) What is the probability of getting a total of 7?

Frequently Asked Questions (FAQs) on Probability

What is probability give an example, what is the formula of probability, what are the different types of probability, what are the basic rules of probability, what is the complement rule in probability.

In probability, the complement rule states that “the sum of probabilities of an event and its complement should be equal to 1”. If A is an event, then the complement rule is given as: P(A) + P(A’) = 1.

What are the different ways to present the probability value?

The three ways to present the probability values are:

  • Decimal or fraction

What does the probability of 0 represent?

The probability of 0 represents that the event will not happen or that it is an impossible event.

What is the sample space for tossing two coins?

The sample space for tossing two coins is: S = {HH, HT, TH, TT}

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Thank you for your best information on probablity

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I really appreciated your explanations because it’s well understandable Thanks

Love the way you teach

There are 3 boxes Box A contains 10 bulbs out of which 4 are dead box b contains 6 bulbs out of which 1 is dead box c contains 8 bulbs out of which 3 are dead. If a dead bulb is picked at random find the probability that it is from which box?

Probability of selecting a dead bulb from the first box = (1/3) x (4/10) = 4/30 Probability of selecting a dead bulb from the second box = (1/3) x (1/6) = 1/18 Probability of selecting a dead bulb from the third box = (1/3) x (3/8) = 3/24 = 1/8 Total probability = (4/30) + (1/18) + (1/8) = (48 + 20 + 45)360 =113/360

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Probability

How likely something is to happen.

Many events can't be predicted with total certainty. The best we can say is how likely they are to happen, using the idea of probability.

Tossing a Coin

When a coin is tossed, there are two possible outcomes:

Heads (H) or Tails (T)

  • the probability of the coin landing H is ½
  • the probability of the coin landing T is ½

Throwing Dice

When a single die is thrown, there are six possible outcomes: 1, 2, 3, 4, 5, 6 .

The probability of any one of them is 1 6

In general:

Probability of an event happening = Number of ways it can happen Total number of outcomes

Example: the chances of rolling a "4" with a die

Number of ways it can happen: 1 (there is only 1 face with a "4" on it)

Total number of outcomes: 6 (there are 6 faces altogether)

So the probability = 1 6

Example: there are 5 marbles in a bag: 4 are blue, and 1 is red. What is the probability that a blue marble gets picked?

Number of ways it can happen: 4 (there are 4 blues)

Total number of outcomes: 5 (there are 5 marbles in total)

So the probability = 4 5 = 0.8

Probability Line

We can show probability on a Probability Line :

Probability is always between 0 and 1

Probability is Just a Guide

Probability does not tell us exactly what will happen, it is just a guide

Example: toss a coin 100 times, how many Heads will come up?

Probability says that heads have a ½ chance, so we can expect 50 Heads .

But when we actually try it we might get 48 heads, or 55 heads ... or anything really, but in most cases it will be a number near 50.

Learn more at Probability Index .

Some words have special meaning in Probability:

Experiment : a repeatable procedure with a set of possible results.

Example: Throwing dice

We can throw the dice again and again, so it is repeatable.

The set of possible results from any single throw is {1, 2, 3, 4, 5, 6}

Outcome: A possible result.

Example: "6" is one of the outcomes of a throw of a die.

Trial: A single performance of an experiment.

Example: I conducted a coin toss experiment. After 4 trials I got these results:

Three trials had the outcome "Head", and one trial had the outcome "Tail"

Sample Space: all the possible outcomes of an experiment.

Example: choosing a card from a deck

There are 52 cards in a deck (not including Jokers)

So the Sample Space is all 52 possible cards : {Ace of Hearts, 2 of Hearts, etc... }

The Sample Space is made up of Sample Points:

Sample Point: just one of the possible outcomes

Example: Deck of Cards

  • the 5 of Clubs is a sample point
  • the King of Hearts is a sample point

"King" is not a sample point. There are 4 Kings, so that is 4 different sample points.

There are 6 different sample points in that sample space.

Event: one or more outcomes of an experiment

Example Events:

An event can be just one outcome:

  • Getting a Tail when tossing a coin
  • Rolling a "5"

An event can include more than one outcome:

  • Choosing a "King" from a deck of cards (any of the 4 Kings)
  • Rolling an "even number" (2, 4 or 6)

Hey, let's use those words, so you get used to them:

Example: Alex wants to see how many times a "double" comes up when throwing 2 dice.

The Sample Space is all possible Outcomes (36 Sample Points):

{1,1} {1,2} {1,3} {1,4} ... ... ... {6,3} {6,4} {6,5} {6,6}

The Event Alex is looking for is a "double", where both dice have the same number. It is made up of these 6 Sample Points :

{1,1} {2,2} {3,3} {4,4} {5,5} and {6,6}

These are Alex's Results:

 After 100 Trials , Alex has 19 "double" Events ... is that close to what you would expect?

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  • Practice Test Questions

Basic Probability and Statistics Quick Review and Practice Questions

  • Posted by Brian Stocker
  • Date August 15, 2017
  • Comments 5 comments

Simple Probability and Statistics – A Quick Review

The probability of an event is given by –

The Number Of Ways Event A Can Occur The total number Of Possible Outcomes

So for example if there are 4 red balls and 3 yellow balls in a bag, the probability of choosing a red ball will be  4/7

Another example:

In a certain game, players toss a coin and roll a dice. A player wins if the coin comes up heads, or the dice with a number greater than 4. In 20 games, how many times will a player win?

a. 13 b. 8 c. 11 d. 15

Correct Answer: A

First determine the possible number of outcomes, the sample space of this event will be:

S = { (H,1),(H,2),(H,3),(H,4),(H,5),(H,6) (T,1),(T,2),(T,3),(T,4),(T,5),(T,6) }

So there are a total of 12 outcomes and 8 winning outcomes.  The probability of a win in a single event is P (W)

= 8/12 = 2/3. In 20 games the probability of a win = 2/3 × 20 = 13

probability math test answers

Basic Probability and Statistics Practice Questions

1. There are 3 blue, 1 white and 4 red identical balls inside a bag. If it is aimed to take two balls out of the bag consecutively, what is the probability to have 1 blue and 1 white ball?

a. 3/28 b. 1/12 c. 1/7 d. 3/7

2. A boy has 4 red, 5 green and 2 yellow balls. He  chooses two balls  randomly for play. What is the probability  that one is red and other is green?

a. 2/11 b. 19/22 c. 20/121 d. 9/11

3. There are 5 blue, 5 green and 5 red books on a shelf.  Two books are selected randomly. What is the probability  of choosing two books of different colors?

a. 1/3 b. 2/5 c. 4/7 d. 5/7

4. How many different ways can a reader choose 3 books out of 4, ignoring the order of selection?

a. 3 b. 4 c. 9 d. 12

5. There is a die and a coin. The dice is rolled and the coin is flipped according to the number the die is rolled. If the die is rolled only once, what is the probability of 4 successive heads?

a. 3/64 b. 1/16 c. 3/16 d. 1/4

6. Smith and Simon are playing a card game. Smith will win if the drawn card form the deck of 52 is either 7 or a diamond, and Simon will win if the drawn card is an even number. Which statement is more likely to be correct?

a. Smith will win more games. b. Simon will win more games. c. They have same winning probability. d. Decision could not be made from the provided data.

7. A box contains 30 red, green and blue balls. The probability of drawing a red ball is twice the other colors due to its size. The number of green balls are 3 more than twice the number of blue balls, and blue are 5 less  than the twice the red. What is the probability that 1 st  two balls drawn from the box randomly will be red? 

a. 10/102 b. 11/102 c. 1/29 d. 1/30

8. Sarah has two children and we know that she has a daughter. What is the probability that the other child is a girl as well?

a. 1/4 b. 1/3 c. 1/2 d. 1

Basic Probability and Statistics Answer Key

1. A There are 8 balls in the bag in total. It is important that two balls are taken out of the bag one by one. We can first take the blue then the white, or first white, then the blue. So, we will have two possibilities to be summed up. Since the balls are taken consecutively, we should be careful with the total number of balls for each case:

First blue, then white ball: There are 3 blue balls; so, having a blue ball is 3/8 possible. Then, we have 7 balls left in the bag. The possibility to have a white ball is 1/7.

P = (3/8) * (1/7) = 3/56

First white, then blue ball: There is only 1 white ball; so, having a white ball is 1/8 possible. Then, we have 7 balls left in the bag. The possibility to have a blue ball is 3/7.

P = (1/8) * (3/7) = 3/56

Overall probability is: 3/56 + 3/56 = 3/28

2. A Probability that the 1st ball is red: 4/11 Probability the 2nd ball is green: 5/10 Combined probability is 4/11 * 5/10 = 20/110 = 2/11

3. D Assume that the first book chosen is red. Since we need to choose the second book in green or blue, there are 10 possible books to be chosen out of 15 – 1(that is the red book chosen first) = 14 books. There are equal number of books in each color, so the results will be the same if we think that blue or green book is the first book.

So, the probability will be 10/14 = 5/7.

4. B Ignoring the order means this is a combination problem, not permutation. The reader will choose 3 books out of 4. So, C(4, 3) = 4! / (3! * (4 – 3)!) = 4! / (3! * 1!) = 4

There are 4 different ways. Ignoring the order means this is a combination problem, not permutation. The reader will choose 3 books out of 4. So,

C(4, 3) = 4! / (3! * (4 – 3)!) = 4! / (3! * 1!) = 4

There are 4 different ways.

5. A If the die is rolled for once, it can be 4, 5 or 6 since we are searching for 4 successive heads. We need to think each case separately. There are two possibilities for a coin; heads (H) or tails (T), each possibility of 1/2; we are searching for H. The possibility for a number to appear on the top of the die is 1/6. Die and coin cases are disjoint events. Also, each flip of coin is independent from the other:

Die: 4 coin: HHHH : 1 permutation P = (1/6) * (1/2) * (1/2) * (1/2) * (1/2) = (1/6) * (1/16)

Die: 5 coin: HHHHT, THHHH, HHHHH : 3 permutations P = (1/6) * 3 * (1/2) * (1/2) * (1/2) * (1/2) * (1/2) = (1/6) * (3/32)

Die: 6 coin: HHHHTT, TTHHHH, THHHHT, HHHHHT, THHHHH, HTHHHH, HHHHTH, HHHHHH : 8 permutations P = (1/6) * 8 * (1/2) * (1/2) * (1/2) * (1/2) * (1/2) * (1/2) = (1/6) * (8/64)

The overall probability is: Pall = (1/6) * (1/16) + (1/6) * (3/32) + (1/6) * (8/64) = (1/6) * (1/16 + 3/32 + 8/64) = (1/6) * (4 + 6 + 8) / 64 = (1/6) * (18/64) = 3/64

6. B There are 52 cards in total. If we closely observe Smith has  16 cards in which he can win. So his winning probability in a single game will be 16/52 on the other hand Simon has 20 cards of wining so his probability on win in single draw is 20/52.

7. A Let the number of red balls be x Then number of blue balls = 2x – 5 Then number of green balls= 2(2x – 5) + 3 = 4x – 10 + 3

= 4x – 7

As there are total 30 balls so the equation becomes x + 2x – 5 + 4x – 7 = 30 x = 6 Red balls are 6, blue are 7 and green are 17. As the probability of drawing a red ball is twice than the others, let’s take them as 12. So the total number of balls will be 36.

Probability of drawing the 1st red: 12/36 Probability of drawing the 2nd red: 10/34 Combined probability = 12/36 X 10/34 = 10/102

8. B At first glance; we can think that a child can be either a girl or a boy, so the probability for the other child to be a girl is 1/2. However, we need to think deeper. The combinations of two children can be as follows:

boy + girl boy + boy girl + boy girl + girl So, the sample space is S = {BG, BB, GB, GG} where the sequence is important. Sarah has a girl; this is the fact. So, calling this as event A, here are the possibilities:

boy + girl girl + boy girl + girl

We eliminate boy + boy, since one child is a girl. A = {BG, GB, GG} The event that Sarah has two girls: B = {GG} We need to compute: P(B|A. = P(B ∩ A. / P(A. = 1/3

More Statistics Practice

Mean, Median and Mode

Making Inferences from a Sample

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Hi there – I will be back to read much more

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The answer to number 2 should be 4/11, which isn’t an option. The wording implies that the order in which the balls are picked out doesn’t matter. The balls can be picked out as red then green, or green then red.

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The answer is correct – Combined probability is 4/11 * 5/10 = 20/110 = 10/55 = 2/11

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I don’t get it, in #7, why can you take the number of red balls as 12?

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4th is confusing. i need more explanation pls.

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Probability

Probability practice test question answers (sample worksheet pdf).

Probability Practice Test Question Answers 2020

Probability Practice Test Question Answers (Sample Worksheet PDF): The situation that may or may not happen, have a chance of happening. The probability of an event that is certain to happen is ‘1’. The probability of an event that is impossible to happen is ‘0’.  The probability of an event =   Test Name Probability …

Probability

Revision Notes

Probability describes how likely something is to happen, it can be written as a fraction, a decimal or as a percentage.

Writing Probability and The Probability Scale

The probability scale runs from 0 to 1

probability math test answers

0 represents something that is impossible If something has a probability of 1 it is certain to happen

Here we have a fair spinner. (We say the spinner is fair if the chance of landing on each side is equal.)

probability math test answers

There are 4 sides, one of the sides has an A on it. We can say the probability of the spinner landing on A is 1 ⁄ 4 (or 0.25 or 25%)

If we mark this on the probability scale, it will be 1 ⁄ 4 of the way along the scale.

probability math test answers

The probability of the spinner landing on C is also 1 ⁄ 4

probability math test answers

2 of the 4 sides have a B on. We can say the probability of the spinner landing on B is 2 ⁄ 4 which can be simplified to 1 ⁄ 2 (or 0.5 or 50%)

If we mark this on the probability scale, it will be 1 ⁄ 2 of the way along the scale.

probability math test answers

The probability of getting any letter other than A, B or C is 0. It would be impossible for the spinner to land on D. We can mark D on the probability scale at 0.

probability math test answers

Example 1: In a bag there are 5 red counters, 4 blue counters and 3 yellow counters A counter is picked at random find the probability that the counter will be a) Red b) Blue c) Yellow

The total number of counters in the bag = 5 + 4 + 3 = 12

a) 5 out of 12 counters are red the probability of picking a red counter will be 5 ⁄ 12 b) 4 out of 12 counters are blue, the probability of picking a blue counter will be 4 ⁄ 12 c) 3 out of 12 counters are yellow, the probability of picking a yellow counter will be 3 ⁄ 12

5 ⁄ 12 + 4 ⁄ 12 + 3 ⁄ 12 = 1

Because the counter can only be red or blue or yellow the three probabilities must add up to 1

The probabilities of all possible events will always add to 1 (or 100%)

Example 2: The probability of it raining tomorrow is 60%. Work out the probability of it not raining tomorrow.

The probability of it raining and the probability of it not raining must add up to 1 (or 100%)

100% - 60% = 40%

The probability of it not raining = 40%

Example 3: The probability of Roger winning a tennis match is 7 ⁄ 12 . Work out the probability of Roger not winning the tennis match.

The probability of Roger winning and the probability of Roger not winning must add up to 1

1 - 7 ⁄ 12 = 5 ⁄ 12

The probability of Roger not winning = 5 ⁄ 12

Example 4: There are 20 pens in a box. 8 of the pens are black 3 of the pens are green. The rest of the pens are red.

One of the pens is chosen at random Find the probability that the pen is red.

We can work out how many red pens there are by calculating 20 - 8 - 3 20 - 8 - 3 = 9

There are 9 red pens and there are 20 pens in total

The probability of getting a red pen = 9 ⁄ 20

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