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# Independent and Dependent Events Study Guide

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## Introduction to Independent and Dependent Events

To us probability is the very guide of life.

—BISHOP JOSEPH BUTLER (1692–1752)

This lesson will explain the difference between independent and dependent events, and provide examples of word problems on these topics.

### Independent Events

Independent events are a set of events where the probability of the second event does not depend on the outcome of the first. In this situation, the probabilities for each individual event are multiplied together.

#### Tip:

The formula to find the probability of two independent events A and B is P(A and B) = P(A) × P(B). The key word and tells you to multiply the probability for each event to find the probability that both events will happen.

Take the following situation:

A jar contains 4 blue marbles, 3 red marbles, and 1 green marble. One marble is selected, then replaced, and then a second marble is chosen. Find the probability for each situation.

What is the probability that a blue marble is selected, and then a green marble is selected?

Read and understand the question. This question is looking for the probability of selecting a blue marble first, and then a green marble second. The events are independent because the first marble is replaced.

Make a plan. Use the facts that there are 4 blue marbles and 1 green marble, and there are a total of 8 marbles. Find the probability of each event and multiply them.

Carry out the plan. The probability of selecting a blue marble at random is , and the probability of selecting a green marble at random is . Multiply these probabilities to find the probability that both events will happen: .

Check your answer. To check this solution, multiply the probabilities again. Because , this solution is checking.

What is the probability that two blue marbles are selected?

Read and understand the question. This question is looking for the probability of selecting a blue marble first, and then another blue marble second. The events are independent because the first marble is replaced.

Make a plan. Use the facts that there are 4 blue marbles, and there are a total of 8 marbles. Find the probability of each event and multiply them.

Carry out the plan. The probability of selecting a blue marble at random is . Multiply the probability of drawing a blue marble by itself to find the probability that both events will happen: .

Check your answer. To check this solution, multiply the probabilities together again. Because , this solution is checking.

#### Tip:

Problems with replacement are independent events.

## Dependent Events

Dependent events are events where the probability of the event following another depends upon the outcome of the previous event. In other words, the likelihood of an event is determined by what is selected first. These problems are commonly known as dependent events because the second event depends on the first.

#### Tip:

The formula to find the probability of two dependent events A and B is also P(A and B) = P(A) × P(B). The key word and tells you to multiply the probability for each event together to find the probability that both events happen. However, the probability of the second event will depend upon the outcome of the first event, so be mindful of this fact as you calculate the chances of each event.

Now, let's use the same situation mentioned earlier in this lesson, except this time, the first , marble is not replaced.

A jar contains 4 blue marbles, 3 red marbles, and 1 green marble. One marble is selected, not replaced, and then a second marble is chosen. Find the probability for each situation.

What is the probability that a blue marble is selected, and then a green marble is selected?

Read and understand the question. This question is looking for the probability of selecting a blue marble first, and a green marble second. The events are dependent events because the first marble is not replaced after it is selected.

Make a plan. Use the facts that there are 4 blue marbles and 1 green marble, and there are a total of 8 marbles. Because the first marble is not replaced after it is chosen, the number of marbles decreases by one for the second draw. Find the probability of each event and multiply them.

Carry out the plan. The probability of first selecting a blue marble at random is , which leaves only 7 marbles left. The probability of selecting a green marble second is . Multiply these probabilities to find the probability that both events will happen: .

Check your answer. To check this solution, multiply the probabilities again. Because , this answer is checking.

What is the probability that two blue marbles are selected?

Read and understand the question. This question is looking for the probability of selecting a blue marble first, and another blue marble second. The events are dependent events because the first marble is not replaced after it is selected.

Make a plan. Use the facts that there are 4 blue marbles, and there are a total of 8 marbles. Because the first marble is not replaced after it is chosen, the number of marbles decreases by one for the second draw. Find the probability of each event and multiply them together.

Carry out the plan. The probability of first selecting a blue marble at random is , which leaves only 7 marbles. The probability of selecting another blue marble second is . Multiply these probabilities to find the probability that both events will happen: .

Check your answer. To check this solution, multiply the probabilities together again. Because , this solution is checking.

What is the probability that two green marbles are selected?

Read and understand the question. This question is looking for the probability of selecting a green marble first, and another green marble second. The events are dependent events because the first marble is not replaced after it is selected.

Make a plan. Use the facts that there is only 1 green marble, and there are a total of 8 marbles. Because the first marble is not replaced after it is chosen, the number of marbles decreases by one for the second draw. Find the probability of each event and multiply them.

Carry out the plan. The probability of first selecting a green marble at random is , which leaves only 7 marbles, none of which are green. Thus, the probability of selecting a green marble second is . Multiply these probabilities to find the probability that both events will happen: = 0.

Check your answer. To check this solution, use the fact that there is only one green marble. The probability of selecting two green marbles without replacing the first is an impossible event. The probability of any impossible event is equal to 0. This answer is checking.

#### Tip:

Problems without replacement are dependent events.

Find practice problems and solutions for these concepts at Independent and Dependent Events Practice Questions.

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