# Independent Events: Definition, Formula & Examples

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Lesson Transcript
Instructor: Eric Istre

Eric has taught high school mathematics for more than 20 years and has a master's degree in educational administration.

This lesson will define independent events, give several scenarios in which they occur, provide the formula for finding the probability of these types of events, and explain how to use the formula.

## What Are Independent Events?

When you become an adult, one of the first things that you really enjoy is that feeling of independence. You are able to make your own decisions. Being independent means that the decisions of others no longer have direct rule over you. Independent events in mathematics are similar to this idea.

Independent events do not affect one another's probability of occurring. For example, if I roll a standard six-sided die and flip a coin, the two events will not have any effect on the probability of the other. Regardless of the outcome of rolling the die, the coin will be just as likely to land on heads or tails. Likewise, regardless of the outcome of the coin flip, the die will be just as likely to land on one of the six numbers of the die.

Here is the formula for finding the probability of independent events A and B.

P(A and B) = P(A) * P(B)

P(A and B) means the probability of A and B both occurring is called a compound event.

P(A) means the probability of A occurring.

P(B) means the probability of B occurring.

A probability can be written in decimal form or in fractional form. If it is written in fractional form, the numerator is the number of successful outcomes, while the denominator is the number of total outcomes.

The formula shows that the probability of each individual event must be determined. Then, the probabilities of the two individual events are multiplied together. This process can actually be used for more than two independent events as well. The examples in this lesson will only discuss two independent events.

## Rolling a Die and Flipping a Coin

Let's go back to the person who is rolling a die and flipping a coin. What is the probability of rolling a number less than 5 and getting tails? We can write this out as:

P(<5 and T), which we know equals P(<5 * PT), then we can plug in the numbers. There are four numbers less than five, so that would go in the numerator of the first part. There are six sides on a die, so that goes on the bottom. So the first part of the equation is 4/6, then for the coin, we know that there is only one possible outcome we're looking for out of two possible options: heads or tails. So we can write that as 1/2. Now we multiply those numbers together to get 2/6, or 1/3.

This means that if you repeat the compound event three times, rolling the die and flipping the coin, you are likely to get a number less than 5 and tails one of those times. This is not a guarantee. This is referred to as theoretical probability: the mathematically expected outcome. What actually happens, experimental probability, can be different than what you expect.

## Drawing Cards from a Standard Deck

Suppose you have a standard deck of playing cards. This means that there are four suits (two black and two red): clubs and spades (black) and hearts and diamonds (red). Each suit has the following cards: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King. What would be the probability of drawing two cards and having them both be red with replacement?

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