Probability Formula - Math Steps, Examples & Questions

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Probability formula

Here you will learn about probability formula, including mutually exclusive events, independent events and conditional probability.

Students will first learn about probability formulas as part of statistics and probability in high school.

What is probability formula?

Probability formulas are used to calculate the probability of single and multiple events.

Probabilities are values that show the likelihood of an event. There are many different types of probability.

Finding the basic probability of event $A$ happening can be calculated using the formula

\text{P(A)}=\cfrac{\text{Number of times A occurs }}{\text{Total number of possible outcomes}},

where $P(A)$ is the notation used to mean “the probability of $A$ happening”.

If the probability needed is more complicated and involves multiple events, consider if the events satisfy conditions such as being mutually exclusive or independent.

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Probability of an event not happening

The total probability of all events happening is $1$ or $100 \%.$

To calculate the probability of an event not happening, use the formula

P(\text{not } A)=1-P(A).

This can also be written as

P\left(A^{\prime}\right)=1-P(A),

where $A’$ refers to the complement of $A$ or $“$ not $A”.$

For example,

The probability of it raining on Monday is $0.3.$

What is the probability of it not raining on Monday?

$P$ (not raining on Monday) $=1-0.3=0.7$

Mutually exclusive events

Mutually exclusive events can be thought of as disjoint events. They cannot happen at the same time.

For example,

When rolling a single fair die, rolling an odd number is mutually exclusive to rolling an even number.

To calculate the probability of either mutually exclusive event occurring, $A$ and $B,$ use the formula

P(A\text{ or }B)=P(A)+P(B).

For example,

The probability of rolling a $2$ or an odd number on a fair six-sided die,

P(2\text{ or odd})=P(2)+P(\text{odd})=\cfrac{1}{6}+\cfrac{3}{6}=\cfrac{4}{6}.

Since $2$ is an even number and not odd, these two events are mutually exclusive.

However, to calculate the probability of rolling an odd number or a prime number, these events are not mutually exclusive so you cannot add the individual probabilities.

The numbers $3$ and $5$ are odd and prime.

To calculate the probability of rolling an odd number or a prime number, it helps to use a sample space diagram or a Venn diagram.

For example,

The Venn diagram shows the two sets, odd numbers and prime numbers.

To calculate P(odd or prime), you need the union of the two sets.

The union of two sets is shown using the $\cup$ symbol, this has been shaded in the Venn diagram below.

P(\text{ Odd }\cup\text{ Prime })=\cfrac{4}{6}

Mutually exclusive events can also be displayed in a probability distribution.

For example,

This probability distribution shows the probabilities of a $4$ -sided spinner landing on the numbers $1$ to $4.$

The probabilities listed in a probability distribution can be based on theoretical probability or experimental probabilities that come from calculating the relative frequency of an event occurring during an experiment.

The formula for relative frequency is

Relative frequency $=\cfrac{\text{frequency of event occurring}}{\text{total number of trials of the experiment}}.$

Step-by-step guide: Mutually exclusive events

Independent events

Independent events are events which are not affected by the occurrence of other events.

If you 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, $P(A\text{ and }B)=P(A)\times{P(B)}$

For example,

The probability of flipping a coin twice and it landing on heads both times,

P(\text {Head and Head})=P(\text{Head}) \times P(\text{Head})=\cfrac{1}{2}\times \cfrac{1}{2}=\cfrac{1}{4}.

Step-by-step guide: Independent events

Dependent events

When events are dependent, you cannot multiply the probability of two separate events together. Instead, consider the probability of the second event given that the first event happened. This is known as conditional probability.

To calculate probability of event $B$ given event $A,$ use Venn diagrams or tree diagrams.

For example,

Charlie has a bag of apples. There are $5$ yellow apples and $4$ red apples.

Charlie is going to take an apple at random, eat it and then take a second apple to eat.

To calculate the probability of Charlie eating an apple of each color, display the probabilities on a tree diagram.

You can see that the probabilities on the second set of branches are different from the probabilities on the first set and to each other. This is because the probabilities of choosing the second apple depend on the color of the previous apple.

Step-by-step guide: Conditional probability

Other probability formulas

The addition rule of probability, says that $P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B).$

For example,

The Venn diagram shows the number of students in a college that study math, English, neither or both.

\begin{aligned}&P(\text{Math or English}) \\\\ &=P(\text{Math})+P(\text{English})-P(\text{Math and English}) \\\\ &=\cfrac{32}{53}+\cfrac{29}{53}-\cfrac{12}{53} \\\\ &=\cfrac{49}{53} \end{aligned}

The multiplication rule says that $P(A\text{ and }B)=P(A)P(B\mid{A}).$ This is read as the probability of $A$ and $B$ is equal to the probability of $A$ occurring times the probability of $B$ occuring, given that $A$ has occurred.

For example,

You can calculate the probability of choosing a student who is only studying math and then a student who is only studying English.

\begin{aligned}&P(\text{Only Math and only English}) \\\\ &=P(\text{Only Math})P(\text{Only English}\mid\text{Only Math}) \\\\ &=\cfrac{20}{53}\cdot\cfrac{17}{52} \\\\ &=\cfrac{340}{2,756} \\\\ &=0.123\approx12 \% \end{aligned}

What is probability formula?

Common Core State Standards

How does this relate to high school math?

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.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, $P(A \text { and } B)=P(A) P(B \mid A)=P(B)P(A \mid B),$ and interpret the answer in terms of the model.

How to use probability formula

In order to use a probability formula:

Decide if the probability is for a single event or multiple events.
Choose the correct formula to use.
Calculate the probability as a fraction, decimal or percentage.

Probability formula examples

Example 1: probability of a single event happening

Each letter of the alphabet is written on a card and the cards are shuffled.

Find the probability of randomly selecting a vowel.

Decide if the probability is for a single event or multiple events.

This represents a single event.

2Choose the correct formula to use.

Use the basic probability formula

P(A)=\cfrac{\text { Number of times event } A \text { occurs }}{\text { Total number of possible outcomes }}.

3Calculate the probability as a fraction, decimal or percentage.

There are $5$ vowels in the alphabet.

P(\text { vowel })=\cfrac{5}{26}

Example 2: probability of not A

The probability of a bus being late on Tuesday is $0.15.$

What is the probability of the bus not being late?

Decide if the probability is for a single event or multiple events.

This represents a single event.

Choose the correct formula to use.

Use the formula for $P(\text{not } A).$

$P\left(A^{\prime}\right)=1-P(A)$

Calculate the probability as a fraction, decimal or percentage.

\begin{aligned}& P(\text { late })=0.15 \\\\ & P(\text { not late })=1-0.15=0.85 \end{aligned}

Example 3: probability of event A or B happening

A fair dice is rolled. Find the probability of it landing on a $2$ or a $3.$

Decide if the probability is for a single event or multiple events.

This represents two events, either landing on a $2$ or a $3.$

Choose the correct formula to use.

The two events are mutually exclusive, use

$P(A \text { or } B)=P(A)+P(B).$

Calculate the probability as a fraction, decimal or percentage.

\begin{aligned}P(2 \text { or } 3) & =P(2)+P(3) \\\\ & =\cfrac{1}{6}+\cfrac{1}{6} \\\\ & =\cfrac{2}{6} \end{aligned}

Example 4: probability of event A and B happening

Kelly plays a game of pickleball and a game of ping pong. The probability that she wins pickleball is $0.7.$ The probability that she wins ping pong is $0.6$

Find the probability that she wins both games.

Decide if the probability is for a single event or multiple events.

This represents two events, winning pickleball and winning ping pong.

Choose the correct formula to use.

The two events are independent, so use

$P(A \text { and } B)=P(A) \times P(B).$

Calculate the probability as a fraction, decimal or percentage.

\begin{aligned}P(\text { wins pickleball and wins ping pong }) & =P(\text { wins pickleball }) \times P(\text { wins ping pong }) \\\\ & =0.7 \times 0.6 \\\\ & =0.42 \end{aligned}

Example 5: multiplication rule

The two-way table shows information about sophomore and junior students studying languages.

A student is selected at random. Find the probability of choosing a junior who studies French after choosing a sophomore who studies German.

Decide if the probability is for a single event or multiple events.

This represents two events, being a junior who studies French and a sophomore who studies German.

Choose the correct formula to use.

This is a conditional probability, so use the formula $P(A\text{ and }B)=P(A)P(B\mid{A}).$ This is read as the probability of $A$ and $B$ is equal to the probability of $A$ occurring times the probability of $B$ occuring, given that $A$ has occurred.

Calculate the probability as a fraction, decimal or percentage.

\begin{aligned}& P \text { (Junior and French) } \\\\ & =P\text { (Junior French)P(Sophomore German|Junior French) } \\\\ & =\cfrac{20}{90} \cdot \cfrac{21}{89} \\\\ & =\cfrac{420}{8,010} \\\\ & =0.052\approx5\% \end{aligned}

Example 6: addition rule

A card is drawn from a standard deck. There are $52$ cards in the deck of cards. What is the probability that the card drawn is either a $7$ or a face card?

Decide if the probability is for a single event or multiple events.

This represents two events, either drawing a $7$ or a face card.

Choose the correct formula to use.

The two events are mutually exclusive, use

$P(A \text { or } B)=P(A)+P(B).$

Calculate the probability as a fraction, decimal or percentage.

\begin{aligned}P(7 \text { or } \text { face card })&=P(7)+P(\text { face card }) \\\\ &=\cfrac{4}{52}+\cfrac{12}{52} \\\\ &=\cfrac{16}{52} \\\\ &=\cfrac{4}{13} \end{aligned}

Teaching tips for probability formula

Introduce probability theory with real life examples that are easy to replicate in the classroom, like with a deck of playing cards. This way students can solve probability problems with a strategy, besides the formula, to check their work.

When teaching probability, explain a random experiment as an action with uncertain outcomes, like rolling a die. Show how a random variable represents each outcome numerically, and use a subset to group specific outcomes, such as the subset of rolling an even number. This approach clarifies how events, outcomes, and probabilities connect.

Easy mistakes to make

Calculating probabilities greater than $\bf{1}$
The total probability of all possible events will add to $1.$ If you find a probability for combined events and get an answer greater than one, check the mistake(s) in your method.

Adding probabilities instead of multiplying them
Remember the sum of probabilities is for “or” events. The probability of this OR that happening. The product of probabilities is for “and” events. The probability of this AND that happening.

Not changing the probability for the second pick when picking two objects
If a bag contains $6$ milk chocolates and $6$ dark chocolates. The probability of selecting a milk chocolate at random is $\cfrac{6}{12}$ or $\cfrac{1}{2}.$

If that chocolate is eaten and another selected, the probability of choosing another milk chocolate is no longer $\cfrac{6}{12}$ or $\cfrac{1}{2},$ it is now $\cfrac{5}{11}.$

Practice probability formula questions

P(A) \times P(B)

P(A)+P(B)

\cfrac{\text{frequency of event occurring}}{\text{total number of trials of the experiment}}

\cfrac{\text{Number of times A occurs }}{\text{Total number of possible outcomes}}

To find the probability of $A$ or $B$ when $A$ and $B$ are mutually exclusive events you add the two probabilities.

\cfrac{3}{5}

\cfrac{2}{5}

\cfrac{2}{25}

\cfrac{2}{4}

Landing on $A$ or $C$ cannot happen at the same time; they are mutually exclusive. To find the probability of $A$ or $C,$ add the two individual probabilities.

\cfrac{1}{5}+\cfrac{2}{5}=\cfrac{3}{5}

\cfrac{2}{8}

\cfrac{1}{6}

\cfrac{1}{12}

\cfrac{8}{12}

Flipping a coin and rolling a die are two separate events. The outcome of one, does not affect the others; they are independent events. To find the probability of the compound event, multiply the two independent probabilities.

\cfrac{1}{2}\times \cfrac{1}{6}=\cfrac{1}{12}

0.35

0.65

0.45

0.5

The total probability of all events happening is 1. There are two total events: winning or not winning. So, $P(\text{not }A)=1-P(A).$

P(\text{Dalia not winning})=1-0.65=0.35

\cfrac{20}{80}

\cfrac{25}{80}

\cfrac{30}{80}

\cfrac{20}{30}

We need the fraction of students who do not study math, that study art.

There are $20$ students who only study art and $10$ students who study neither math nor art. This makes $30$ total students do not study math, including the $20$ who study art.

The probability is the ratio of the students who study art over the total students who do not study math:

P(\text { study art } \mid \text { doesn't study math })=\cfrac{20}{30}

\cfrac{33}{49}

\cfrac{2}{5}

\cfrac{1}{10}

\cfrac{27}{49}

Randomly choosing a student who plays basketball or none of the sports cannot happen at the same time; they are mutually exclusive. To find the probability of $B$ or none, add the two individual probabilities.

\cfrac{30}{49}+\cfrac{3}{49}=\cfrac{33}{49}

Probability formula FAQs

What is the ratio used for finding the probability?

The numerator is referred to as the desired outcome, number of favorable outcomes. The denominator is the total number of possible outcomes.

What is the main difference between the normal distribution and the binomial distribution?

The normal distribution is a continuous distribution that describes data with a symmetric, bell-shaped curve. The binomial distribution is discrete and used for counting the number of successes in a fixed number of trials.

How does standard deviation relate to the normal distribution?

In the normal distribution, the standard deviation measures how spread out the values are around the mean.

The next lessons are

Compound probability
Probability distribution
Units of measurement

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