Independent events

Here you will learn about independent events, including what independent events are and how to calculate the probability of independent events occurring.

Students will first learn about independent events as part of statistics and probability in 7 th grade and continue to learn about them in high school.

What are independent events?

Independent events are events whose probability of occurrence is not affected by other events.

For example, if you roll a die twice, the outcome of the first roll and second roll have no effect on each other – they are independent. The chances of rolling a 1, \; 2, \; 3, \; 4, \; 5 or 6 are the same on each roll, as shown in the table.

If two events are independent then P(A \text { and } B)=P(A) \times P(B).

This is called the multiplication rule.

Let’s look at an example.

If you are rolling a dice twice, we can find the probability of getting two sixes.

The probability of getting a 6 is \cfrac{1}{6}.

So the probability of getting a 6 and a 6 is \cfrac{1}{6} \times\cfrac{1}{6} = \cfrac{1}{36}.

What are independent events?

Common Core State Standards

How does this relate to 7 th grade math and 5 th grade math?

• 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.

How to calculate the probability of independent events

In order to calculate the probability of independent events:

1. Confirm that the events are independent.
2. Identify the probabilities of the events.
3. Multiply the probabilities.

Independent event examples

Example 1: identifying independent events

Look at the events below. Which are examples of independent events?

• Rolling a die and getting a 6 and tossing a coin and getting a head.
• Tossing two coins and getting two heads.
• Choosing a green marble from a bag containing blue and yellow marbles and then choosing another green marble from the same bag.
• A student in Mr. Lopez’s class has brown hair and shoe size 6.

The definition of independence is that the events have no effect on each other.

When you roll a die and toss a coin, the outcome of one does not affect the outcome of the other therefore, the events are independent.

When you toss two coins, the outcome of one coin toss does not affect the other therefore, the events are independent.

When you choose a marble from a bag, you are changing the number of marbles in the bag, and therefore, the outcome of choosing the first marble will have an effect on the outcome of the second drawing for the second marble.

For example, if you choose a green marble the first time, the probability of choosing a green marble the second time will decrease. Therefore, these are dependent events.

The color of someone’s hair does not affect the size of their feet therefore, the events are independent.

Example 2: probability of two independent events

The probability that Tom forgets his homework is 0.3. The probability that Noah forgets his homework is 0.1. The events are independent. Calculate the probability that both Tom and Noah forget their homework on the same day.

Example 3: probability of three independent events

Rachel tosses three fair coins. Find the probability that all three coins land on tails.

Example 4: using a tree diagram

Niamh plays a game of ping pong on Saturday and a game of badminton on Sunday. The different outcomes are represented on the tree diagram below.

Calculate the probability that Niamh wins both games.

Example 5: finding the probability of individual events

Zainab has two bags of mixed candy. She picks one candy from each bag. The probability that Zainab picks a lollipop from the first bag is \cfrac{1}{6} and the probability that she picks a lollipop from both bags is \cfrac{1}{15}. What is the probability that she picks a lollipop from the second bag?

Example 6: finding the probability of individual events

Rosie flips a biased coin twice. The probability that Rosie gets two heads is 0.64. Find the probability that Rosie gets a head on a single coin flip.

Teaching tips for independent events

• Let students experience independent events in the classroom, but using examples that involve dice, coins, cards, marbles and other commonly found objects.

• Have students explore the sample spaces of independent events before introducing the rule. Many students will figure out the multiplication rule without being told. And even if they don’t, they will have more than one strategy to utilize when solving.

Easy mistakes to make

• Multiplying or dividing fractions incorrectly
  To multiply fractions, multiply the numerators and multiply the denominators.
  To divide fractions, find the reciprocal of the second fraction and multiply.

• Adding probabilities instead of multiplying them
  For independent events P(A \text { and } B)=P(A) \times P(B).
  For mutually exclusive events P(A \text { or } B)=P(A) + P(B).

Related probability lessons

• Probability
• Probability notation
• How to find probability
• How to calculate probability
• Probability formula
• Mutually exclusive events
• Probability scale
• Conditional probability
• Venn diagram

The next lessons are

• Compound events
• Probability distribution
• Units of measurement

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