How to calculate probability

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

Students will first learn about how to calculate probability as part of statistics and probability in 7 th grade and continue to learn about it in high school.

What is probability?

Probability is the likelihood of an event occurring.

To calculate the probability of an event happening, use the formula

For example,

Let’s look at the probability of getting an even number when a fair 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.

The probability of getting an even number =\cfrac{\text{Number of desired outcomes}}{\text{Total number of outcomes}}=\cfrac{3}{6}

Probabilities range from 0 to 1.

If something has a probability of 0 then it is impossible and if something has a probability of 1 then it is certain. Probabilities can be represented by fractions, decimals or percents.

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

For example,

To write the probability of getting a 1 we could write P(1).

The total set of outcomes is called the sample space. It can be represented by a list, table, tree diagram or other drawing.

For example,

Simple event: The sample space for rolling a fair die is 1, \; 2, \; 3, \; 4, \; 5 and 6.

Compound event: The sample space for a fair coin flipped three times is

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: \textbf{P(A \text {or} B) = P(A) + P(B)}
  
  If you 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 \cfrac{3}{6} and the probability of getting an odd number is \cfrac{3}{6}.
  The probability of getting an even or an odd number is \cfrac{3}{6}+\cfrac{3}{6}=\cfrac{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

• Independent events
  Independent events are events which are not affected by the occurrence of other events. For example, if you roll a die twice, the outcome of the first roll and second roll do not affect each other – they are independent events.
  
  For independent events: \bf {\textbf{P(A \text {and} B) = P(A)} \; \bf{\times} \; \textbf{P(B)}}
  
  Step-by-step guide: Independent events

• 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 color of the first counter was.
  
  Step-by-step guide: Conditional probability

How to calculate probability?

Common Core State Standards

How does this relate to 7 th grade math and high school math?

• Grade 7 – Statistics and Probability (7.SP.C.7)
  Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.

• Grade 7 – Statistics and Probability (7.SP.C.8)
  Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

• Statistics and Probability – Conditional Probability and the rules of Probability (HS.S-CP.A.2)
  Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

• Statistics and Probability – Conditional Probability and the rules of Probability (HS.S-CP.A.3)
  Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

• Statistics and Probability – Conditional Probability and the rules of Probability (HS.S-CP.B.6)
  Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.

• Statistics and Probability – Conditional Probability and the rules of Probability (HS.S-CP.B.7)
  Apply the Addition Rule, P(A \text { or } B) = P(A) + P(B) \; – \; P(A \text { and } B), and interpret the answer in terms of the model.

• Statistics and Probability – Conditional Probability and the rules of Probability (HS.S-CP.B.8)
  Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.

How to calculate probability

In order to calculate probability:

1. Identify the sample space.
2. Write out the basic probability.
3. Solve the problem, using \textbf{AND} or \textbf{OR} rules as appropriate.

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.

1. Identify the sample space.

All possible outcomes are P, \; R, \; O, \; B, \; A, \; B, \; I, \; L, \; I, \; T and Y and each card has an equally likely chance of being chosen.

Note: Because cards B and I repeat, their total probability is higher.

2Write out the basic probability.

Use the formula:

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

3Solve the problem, using \textbf{AND} or \textbf{OR} rules as appropriate.

Not needed for this basic probability question.

Example 2: mutually exclusive events

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

Example 3: independent events

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

Identify the sample space.

Let’s model with a table.

All possible outcomes are shown in the list and each number on the die has an equally likely chance of being chosen.

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.

Solve the problem, using \textbf{AND} or \textbf{OR} rules as appropriate.

You can draw a tree diagram to clearly see the different outcomes.

There are two options for the outcome: 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 or 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.

Identify the sample space.

Let’s model with a list.

List: Red, red, red, red, red, red, blue, blue, blue, blue, blue.

All possible outcomes are shown in the list and each marble has an equally likely chance of being chosen.

Example 6: using a Venn diagram

The Venn diagram below shows the number of students who passed their English and Math classes.

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

Identify the sample space.

The sample space is shown in the highlighted part of Venn-diagram.

Teaching tips for how to calculate probability

• Introduce finding probability with hands-on experiments where students can create their own relative frequency tables and graphs. Start with simple events and then progress to multiple events.

• Let struggling students check their work with a probability calculator or excel program when working independently on probability problems.

Easy mistakes to make

• Confusing the number of events (favorable outcomes) with total possible outcomes
  To calculate probability, always use the number of favorable outcomes as the numerator and the number of total possible outcomes as the denominator.

• Confusing probability rules
  For independent events, use the formula P(A \text { and } B)=P(A) \times(P(B)
  For mutually exclusive events, use the formula P(A \text { 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 marbles and 7 yellow marbles, the probability of picking a blue marbles on the first pick is \cfrac{3}{10} and the probability of picking a yellow marbles on the first pick is \cfrac{7}{10}.
  
  The probabilities of picking the second marble depends on whether the first marble is replaced into the bag or not. This determines which formula to use:
  • Probability of event B and event A is P(A \text { and } B)=P(A) \times(P(B)
  • Probability of even B or event A is P(A \text { or } B)=P(A)+P(B)

Related probability lessons

• Probability
• Probability notation
• How to find probability
• Probability formula
• Probability scale
• Tree diagram probability
• Venn diagram

The next lessons are

• Compound events
• Probability distribution
• Units of measurement

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