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How To Calculate Probability

Here we will learn how to calculate probability, including basic probability, mutually exclusive events, independent events and conditional probability.

There are also calculating probability worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is probability?

Probability is the likelihood of an event occurring.

To find the probability of an event happening we use the formula

For example,

Let’s look at the probability of getting an even number when a die is rolled.

The desired outcome is getting an even number. There are 3 even numbers on a die. 

The total number of possible outcomes is 6 since there are 6 numbers on a die.

Probabilities range from \bf{0} to \bf{1}.

If something has a probability of \bf{0} then it is impossible and if something has a probability of \bf{1} then it is certain .

We use the notation P(event) to represent the probability of an event happening.

If we wanted to write the probability of getting a 1 we could write P (1).

Calculating probabilities of combined events

Sometimes we want to find the probability of more than one event occurring. There are different probability rules that we can use.

• Mutually exclusive events

Mutually exclusive events are two or more events that cannot occur at the same time. For example, getting heads and tails when tossing a coin or rolling a 2 and a 3 on a die.

For mutually exclusive events: P(A or B) = P(A) + P(B)

If we have an exhaustive list of outcomes, their probabilities sum to 1. For example, the probability of getting an even or an odd number on a die.

The probability of getting an even number is \frac{3}{6}

and the probability of getting an odd number is \frac{3}{6}.

The probability of getting an even or an odd number is \frac{3}{6}+\frac{3}{6}=\frac{6}{6}=1.

Since getting an even number or an odd number covers all the possible outcomes, it is an exhaustive list and the probabilities add up to 1.

Step-by-step guide: Mutually exclusive events (coming soon)

• Independent events

Independent events are events which are not affected by the occurrence of other events. For example, if we roll a die twice, the outcome of the first roll and second roll have no effect on each other – they are independent events. 

For independent events: P(A and B) = P(A) x P(B)

Step-by-step guide: Independent events (coming soon)

• Conditional probability

Conditional probability is the probability of an event occurring based on the occurrence of another event.

For conditional probability the probabilities are calculated based on what has already occurred.

For example, there are 5 counters in a bag, 2 are black and the rest are white.

A counter is picked at random and not replaced. A second counter is picked at random. The probability that the second counter is black depends on what the colour of the first counter was.

Step-by-step guide: Conditional probability

How to calculate probability

In order to calculate probability:

• Write out the basic probability
• Solve the problem, using AND or OR rules as appropriate .

Explain how to calculate probability

How to calculate probability worksheet

Get your free how to calculate probability worksheet of 20+ questions and answers. Includes reasoning and applied questions.

How to calculate probability examples

Example 1: basic probability.

Jamie has the following cards,

A card is chosen at random. Find the probability the card has a letter B on it.

Write out the basic probability .

We can write down the basic probability by using

\text{Probability}=\frac{\text{number of desired outcomes}}{\text{total number of outcomes}} .

The number of cards with B is 2, and the total number of cards is 11.

2 Solve the problem using AND or OR rules as appropriate .

Not needed as this is a basic probability question.

Example 2: mutually exclusive events

What is the probability of landing on a 2 or a 3 on the following spinner?

We can write down the probability of getting a 2 and the probability of getting a 3.

P(2)=\frac{3}{8}

P(3)=\frac{2}{8}

Solve the problem using AND or OR rules as appropriate .

The probability of landing on a 2 or a 3 is \frac{5}{8}.

Example 3: independent events

Olivia flips a coin and rolls a die. What is the probability that the coin lands on heads and the die lands on 1 ?

\text{P(1)}=\frac{1}{6}

The probability that the coin lands on heads and the die lands on 1 is \frac{1}{12}.

Example 4: using a tree diagram

The probability that Kate wins a game of tennis is 0.6. The probability that Billy wins a game of tennis is 0.7. Kate plays a match on Saturday and Billy plays a match on Sunday.

Find the probability that one of them wins and one of them loses.

We can work out the probabilities that the people do not win their tennis games.

\text{P(Kate NOT win)}=1-0.6=0.4

\text{P(Billy NOT win)}=1-0.7=0.3

For this question we are going to draw a tree diagram so we can clearly see the different outcomes.

For the outcome that one of them wins and the other loses, we could have Kate wins and Billy loses or Kate loses and Billy wins.

\text{P(Kate wins and Billy loses)}=0.6 \times 0.3=0.18

\text{P(Kate loses and Billy wins)}=0.4 \times 0.7=0.28

\text{P(one wins and one loses)}=0.18+0.28=0.46

The probability that one of them wins and the other loses is 0.46.

Example 5: conditional probability

A bag contains 7 red marbles and 5 blue marbles. 1 marble is chosen at random.

The marble is red. A second marble is chosen. Find the probability the second marble is also red.

These are dependent events. The first event affects the probabilities for the second event.

\text{P(first marble is red)}=\frac{7}{12}

Given that one red marble has been chosen, there are now 6 red marbles and 11 marbles altogether. This is conditional probability.

\text{P(second marble is red)}=\frac{6}{11}

The probability that the second marble is red is \frac{6}{11}.

Example 6: using a Venn diagram

The Venn diagram below shows the number of students who pass their mock exams in English and Maths.

A student is chosen at random. Given that the chosen student passes Maths, find the probability that they did not pass English.

This is a more complex probability question to work out. However we can work out how many students pass Maths.

This is a conditional probability question.

The desired outcome is that the student passes Maths but not English. There are 9 students who pass Maths but not English. The condition is that they pass Maths so we need to consider all the students who pass Maths. We know that 21 students pass Maths.

Therefore the probability that the student did not pass English given that they pass Maths is \frac{9}{21}.

Example 7: using a two-way table

The two-way shows information about the gender and eye colour of the children in class 6.

A child is chosen at random. What is the probability that the child has green eyes given that they are male?

This is a more complex probability question to work out. However we can work out how many children are male.

The desired outcome is that the child has green eyes but is male. There are 2 children who have green eyes and are male. The condition is that they are male, so we need to consider all the children who are male. We know that 15 children are male.

Therefore the probability that the child has green eyes given that they are male is \frac{2}{15}.

Common misconceptions

• Adding probabilities instead of multiplying them

For independent events P(A \ and \ B) = P(A) \times (P(B). For mutually exclusive events P(A \ or \ B) = P(A) + P(B).

• Multiplying or dividing fractions incorrectly

To multiply fractions, multiply the numerators and multiply the denominators. To divide fractions, turn the second one upside down and multiply.

• Adding fractions incorrectly

Remember, fractions can only be added or subtracted if they have a common denominator.

• Not changing the probability for the second pick when picking two objects (conditional probability)

For example, if you have a bag containing 3 blue balls and 7 yellow balls, the probability of picking a blue ball on the first pick is \frac{3}{10} and the probability of picking a yellow ball on the first pick is \frac{7}{10}. The probabilities of picking the second ball depend on whether the first ball is replaced into the bag or not.

Practice how to calculate probability questions

1. Luke has a deck of cards. Luke draws one card at random. Find the probability that Luke picks a King.

There are 4 Kings in a deck of cards. There are 52 cards in total.

2. The probability that Tom wears a certain colour t-shirt is shown below.

Find the probability that Tom wears a black or a grey t-shirt.

3. Evie takes the bus to work. The probability the bus is late on any given day is \frac{2}{5}, independent of whether it was late on the previous day. Find the probability the bus is late two days in a row.

4. Eddy has 10 red socks, 8 blue socks and 2 yellow socks.

Eddy picks a sock from the drawer. It is red.

Eddy picks a second sock. Find the probability it is also red.

Once Eddy has taken a red sock, there will be 9 red socks left and 19 socks left altogether. Therefore the probability is \frac{9}{19}.

5. The two-way table shows information about the height and hair colour of the children in class 3.

A student is chosen at random. Find the probability the student has blonde hair given that they are under 120cm.

The desired outcome is a student that has blonde hair and is under 120cm. There are 6 such students.

The condition is that the student is under 120cm. There are a total of 12 students under 120cm.

The probability is \frac{6}{12}=\frac{1}{2} .

6. Rachel has 9 balls in a bag. 4 of the balls are blue and the other 5 are red. Rachel picks one ball, notes its colour and replaces the ball. She then picks a second ball. Find the probability the two balls she picks are the same colour.

Rachel can pick two red balls or two blue balls.

The total probability is \frac{41}{81} .

How to calculate probability GCSE questions

1. Jason picks one of the following cards at random.

(a) Find the probability that Jason picks a H.

(b) Find the probability that Jason picks an M or an A.

(c) Find the probability that Jason does not pick an M.

(a)  \frac{1}{11}

(b)  \frac{2}{11}+\frac{2}{11}=\frac{4}{11}

(c)  \frac{9}{11}

2. (a) Yasmin designs a game where players must roll a dice and pick a card from a set of cards containing the numbers 1-10 once each. Players win if they roll a multiple of 3 and pick a card that is a multiple of 5.

Find the probability that a player wins the game.

(b) 150 people play the game.

Yasmin charges players £1 to play and winners receive a prize of £5. How much profit should Yasmin expect to make?

Number of winners = \frac{1}{15} \times 150=10

Profit = £150-£50=£100

3. Janet has a bag of sweets containing three strawberry sweets, four blackcurrant sweets and five orange sweets.

Janet picks two sweets at random. Find the probability that both sweets are strawberry.

Learning checklist

You have now learned how to:

• Calculate basic probabilities
• Calculate the probability of two events happening
• Calculate the probability of mutually exclusive events
• Calculate the probability of independent events
• Deal with conditional probability
• Solve probability problems

The next lessons are

• Combined events probability
• Probability distribution
• Describing probability

Still stuck?

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How to Calculate Probability

Last Updated: July 31, 2023 Fact Checked

This article was co-authored by Mario Banuelos, PhD . Mario Banuelos is an Assistant Professor of Mathematics at California State University, Fresno. With over eight years of teaching experience, Mario specializes in mathematical biology, optimization, statistical models for genome evolution, and data science. Mario holds a BA in Mathematics from California State University, Fresno, and a Ph.D. in Applied Mathematics from the University of California, Merced. Mario has taught at both the high school and collegiate levels. There are 8 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 3,372,474 times.

Chances are (pun intended) you've encountered probability by now, but what exactly is probability, and how do you calculate it? Probability is the likelihood of a specific event happening, like winning the lottery or rolling a 6 on a die. Finding probability is easy using the probability formula (the number of favorable outcomes divided by the total number of outcomes). In this article, we'll walk you through exactly how to use the probability formula step by step, plus show you some examples of the probability formula in action.

Finding the Probability of a Single Random Event

Example: It would be impossible to calculate the probability of an event phrased as: “Both a 5 and a 6 will come up on a single roll of a die.”

• Example 1 : What is the likelihood of choosing a day that falls on the weekend when randomly picking a day of the week? "Choosing a day that falls on the weekend" is our event, and the number of outcomes is the total number of days in a week: 7.
• Example 2 : A jar contains 4 blue marbles, 5 red marbles and 11 white marbles. If a marble is drawn from the jar at random, what is the probability that this marble is red? "Choosing a red marble" is our event, and the number of outcomes is the total number of marbles in the jar, 20.

• Example 1 : What is the likelihood of choosing a day that falls on the weekend when randomly picking a day of the week? The number of events is 2 (since 2 days out of the week are weekends), and the number of outcomes is 7. The probability is 2 ÷ 7 = 2/7. You could also express this as 0.285 or 28.5%.
• Example 2 : A jar contains 4 blue marbles, 5 red marbles and 11 white marbles. If a marble is drawn from the jar at random, what is the probability that this marble is red? The number of events is 5 (since there are 5 red marbles), and the number of outcomes is 20. The probability is 5 ÷ 20 = 1/4. You could also express this as 0.25 or 25%.

• For example, the likelihood of rolling a 3 on a 6-sided die is 1/6. But the probability of rolling all five other numbers on a die is also 1/6. 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 6/6 , which = 100%.

Note: If you had, for example, forgotten about the number 4 on the dice, adding up the probabilities would only reach 5/6 or 83%, indicating a problem.

• For example, if you were to calculate the probability of the Easter holiday falling on a Monday in the year 2020, the probability would be 0 because Easter is always on a Sunday.

Calculating the Probability of Multiple Random Events

Note: The probability of the 5s being rolled are called independent events, because what you roll the first time does not affect what happens the second time.

• Now, the likelihood that the second card is a club is 12/51, since 1 club will have already been removed. This is because what you do the first time affects the second. If you draw a 3 of clubs and don't put it back, there will be one less club and one less card in the deck (51 instead of 52).
• The probability that the first marble is red is 5/20, or 1/4. The probability of the second marble being blue is 4/19, since we have 1 less marble, but not 1 less blue marble. And the probability that the third marble is white is 11/18, because we’ve already chosen 2 marbles.

• Example 1 : Two cards are drawn randomly from a deck of cards. What is the likelihood that both cards are clubs? The probability of the first event happening is 13/52. The probability of the second event happening is 12/51. The probability is 13/52 x 12/51 = 12/204 = 1/17. You could also express this as 0.058 or 5.8%.
• Example 2 : A jar contains 4 blue marbles, 5 red marbles and 11 white marbles. If three marbles are drawn from the jar at random, what is the probability that the first marble is red, the second marble is blue, and the third is white? The probability of the first event is 5/20. The probability of the second event is 4/19. And the probability of the third event is 11/18. The probability is 5/20 x 4/19 x 11/18 = 44/1368 = 0.032. You could also express this as 3.2%.

Converting Odds to Probabilities

• The number 11 represents the likelihood of choosing a white marble and the number 9 represents the likelihood of choosing a marble of a different color.
• So, odds are that you will draw a white marble.

• The event that you’ll draw a white marble is 11; the event another color will be drawn is 9. The total number of outcomes is 11 + 9, or 20.

• So, in our example, the probability of drawing a white marble is 11/20. Divide this out: 11 ÷ 20 = 0.55 or 55%.

Probability Cheat Sheets

Expert Q&A

Video . By using this service, some information may be shared with YouTube.

• Mathematicians typically use the term “relative probability” to refer to the chances of an event happening. They insert the word "relative" since no outcome is 100% guaranteed. For example, if you flip a coin 100 times, you probably won't get exactly 50 heads and 50 tails. Relative probability takes this caveat into account. [10] X Research source Thanks Helpful 0 Not Helpful 1
• You may need to know that that in sports betting and bookmaking, odds are expressed as “odds against,” which means that the odds of an event happening are written first, and the odds of an event not happening come second. Although it can be confusing, it's important to know this if you’re planning to bet on a sporting event. Thanks Helpful 12 Not Helpful 4
• The most common ways of writing down probabilities include putting them as fractions, as decimals, as percentages, or on a 1–10 scale. Thanks Helpful 8 Not Helpful 5

You Might Also Like

• ↑ https://www.theproblemsite.com/reference/mathematics/probability/mutually-exclusive-outcomes
• ↑ Mario Banuelos, PhD. Assistant Professor of Mathematics. Expert Interview. 11 December 2021.
• ↑ https://www.mathplanet.com/education/pre-algebra/probability-and-statistic/probability-of-events
• ↑ https://www.mathsisfun.com/probability_line.html
• ↑ https://www.probabilisticworld.com/not-all-zero-probabilities/
• ↑ https://www.khanacademy.org/math/ap-statistics/probability-ap/stats-conditional-probability/e/calculating-conditional-probability
• ↑ https://www.mathsisfun.com/data/probability.html
• ↑ https://www.mathsisfun.com/data/probability-events-types.html

About This Article

Probability is the likelihood that a specific event will occur. To calculate probability, first define the number of possible outcomes that can occur. For example, if someone asks, “What is the probability of choosing a day that falls on the weekend when randomly picking a day of the week,” the number of possible outcomes when choosing a random day of the week is 7, since there are 7 days of the week. Now define the number of events. In this example, the number of events is 2 since 2 days out of the week fall on the weekend. Finally, divide the number of events by the number of outcomes to get the probability. In our example, we would divide 2, the number of events, by 7, the number of outcomes, and get 2/7, or 0.28. You could also express the answer as a percentage, or 28.5%. Therefore, there’s a 28.5% probability that you would choose a day that falls on the weekend when randomly picking a day of the week. To learn how to calculate the probability of multiple events happening in a row, keep reading! Did this summary help you? Yes No

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How to Solve Probability Questions

How to Explain the Sum & Product Rules of Probability

Most probability questions are word problems, which require you to set up the problem and break down the information given to solve. The process to solve the problem is rarely straightforward and takes practice to perfect. Probabilities are used in mathematics and statistics and are found in everyday life, from weather forecasts to sporting events. With a little practice and a few tips, the process of calculating probabilities can be more manageable.

Find the keyword. One important tip when solving a probability word problem is to find the keyword, which helps to identify which rule of probability to use. The keywords are "and," "or" and "not." For instance, consider the following word problem: "What is the probability that Jane will choose both the chocolate and the vanilla ice cream cones given that she chooses chocolate 60 percent of the time, vanilla 70 percent of the time, and neither 10 percent of the time." This problem has the keyword "and."

Find the correct rule of probability. For problems with the keyword "and," the rule of probability to use is a multiplication rule. For problems with the keyword "or," the rule of probability to use is an addition rule. For problems with the keyword "not," the rule of probability to use is the complement rule.

Determine what event is being sought. There may be more than one event. An event is the occurrence in the problem that you are solving the probability for. The example problem is asking for the event that Jane will choose both the chocolate and the vanilla. So in essence, you want the probability of her choosing these two flavors.

Determine whether the events are mutually exclusive or independent if appropriate. When using a rule of multiplication, there are two to choose from. You use the rule P(A and B) = P(A) x P(B) when the events A and B are independent. You use the rule P(A and B) = P(A) x P(B|A) when the events are dependent. P(B|A) is a conditional probability, indicating the probability that event A occurs given that event B has already occurred. Similarly, for the rules of addition, there are two to choose from. You use the rule P(A or B) = P(A) + P(B) if the events are mutually exclusive. You use the rule P(A or B) = P(A) + P(B) - P(A and B) when the events are not mutually exclusive. For the complement rule, you always use the rule P(A) = 1 - P(~A). P(~A) is the probability that event A does not occur.

Find the separate parts of the equation. Each equation of probability has different parts that need to be filled to solve the problem. For the example, you determined the keyword is "and," and the rule to use is a rule of multiplication. Because the events are not dependent, you will use the rule P(A and B) = P(A) x P(B). This step sets P(A) = probability of event A occurring and P(B) = probability of event B occurring. The problem says that P(A = chocolate) = 60% and P(B = vanilla) = 70%.

Substitute the values into the equation. You can substitute the word "chocolate" when you see the event A and the word "vanilla" when you see the event B. Using the appropriate equation for the example and substituting the values, the equation is now P(chocolate and vanilla) = 60% x 70%.

Solve the equation. Using the previous example, P(chocolate and vanilla) = 60 percent x 70 percent. Breaking down the percentages into decimals will yield 0.60 x 0.70, found by dividing both percentages by 100. This multiplication results in the value 0.42. Converting the answer back to a percentage by multiplying by 100 will yield 42 percent.

• Two events are known to be mutually exclusive if they both cannot occur at the same time. If they can occur at the same time, they are not. Two events are known to be independent if one event does not depend on the outcome of the other event. These definitions are used to help complete the previous steps; a working knowledge of these is required to solve these problems.

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About the Author

Michelle Friesen began writing in 2003. Contributing to eHow, she is also a software engineer and adjunct instructor of statistics and computer information systems. Friesen holds a Master of Science in engineering management and a certificate in financial engineering, as well as Bachelor of Science degrees in applied mathematics and computer science from the Missouri University of Science and Technology.

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