Math resources Statistics and probability

Probability

How to calc. prob.

# 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

\text{Probability}=\cfrac{\text{Number of desired outcomes}}{\text{Total number of outcomes}}

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

## 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:

\text{Probability}=\cfrac{\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.

\text{Probability}=\cfrac{\text{Number of desired outcomes}}{\text{Total number of outcomes}}=\cfrac{2}{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?

Identify the sample space.

Write out the basic probability.

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

### 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.

Write out the basic probability

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

### 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.

Identify the sample space.

Write out the basic probability.

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

### 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.

Write out the basic probability.

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

### 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.

Write out the basic probability.

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

### 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)

### Practice how to calculate probability questions

1. Show the sample space of flipping a fair coin and drawing a marble out of the bag.

List – Tails & Blue-1, Tails & Blue-2, Tails & Orange, Tails & Green

Heads or tails AND blue, blue, orange or green

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

In this case, the table is correct, because it has every possible outcome. This includes showing blue twice, since there are two blue marbles.

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

\cfrac{1}{13}

\cfrac{1}{4}

\cfrac{1}{52}

\cfrac{4}{13}

There are 52 cards in total. This is a single event – choosing a King. There are 4 Kings in a deck of cards.

\text { Probability }=\cfrac{\text { desired outcome }}{\text { total possible outcomes }}=\cfrac{4}{52}=\cfrac{1}{13}

3. 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.

\cfrac{9}{19}

\cfrac{10}{20}

\cfrac{9}{20}

\cfrac{10}{19}

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

4. The probability that Tom wears a certain color t-shirt is shown below.

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

0.2

0.15

0.3

0.7
P(A \text{ or } B) = P(A) + P(B)

P(\text{black or gray}) = 0.1+0.2 = 0.3

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

\cfrac{2}{5}

\cfrac{4}{25}

\cfrac{4}{5}

\cfrac{2}{25}
\text{P(A and B)}=P(A) \times P(B)

\text{P(late and late)}=\cfrac{2}{5} \times \cfrac{2}{5} = \cfrac{4}{25}

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

\cfrac{16}{81}

\cfrac{41}{81}

\cfrac{25}{81}

\cfrac{40}{81}

Rachel can pick two red marbles or two blue marbles.

\text{P(red and red)}=\cfrac{4}{9} \times \cfrac{4}{9}=\cfrac{16}{81}

\text{P(blue and blue)}=\cfrac{5}{9} \times \cfrac{5}{9}=\cfrac{25}{81}

\text{P(same color)}=\cfrac{16}{81}+\cfrac{25}{81}=\cfrac{41}{81}

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

## How to calculate probability FAQs

What is the difference between theoretical probability and experimental probability?

Theoretical probability uses math to calculate the probability in theory, while experimental probability reflects the results of an experiment.
For example, when rolling dice the theoretical probability of rolling a 6 is \cfrac{1}{6}.

However, let’s say you conduct an experiment by rolling a dice and you roll a 6 for 20 out of the 100 rolls. Your experimental probability is \cfrac{20}{100} or \cfrac{1}{5}. It is close to the theoretical, but not equal.

What is probability distribution?

The probability distribution shows all possible values or probabilities within a given range. The binomial and normal distributions are commonly used.

What is Bayes’ theorem?

The theorem states a formula for calculating the conditional probability of events.

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