Independent Events: Definition, Formula & Examples

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  • 0:00 What Are Independent Events?
  • 1:42 Rolling a Die and…
  • 2:50 Drawing Cards from a…
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Lesson Transcript
Eric Istre

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

Expert Contributor
Kathryn Boddie

Kathryn earned her Ph.D. in Mathematics from UW-Milwaukee in 2019. She has over 10 years of teaching experience at high school and university level.

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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Additional Activities

Determining if Events are Independent

The probability formula P(A and B) = P(A)*P(B) is used only for events that are independent. If this formula is used on events that are dependent, the result will not actually be the theoretical probability of the events occurring. It is important to be able to determine if events are independent or not so the probability can be calculated correctly.


For each of the following examples, determine if the events are independent or dependent. Support your reasoning.

  • Rolling a standard 6 sided die and then rolling it again
  • Picking two cards out of a standard deck of cards without replacement
  • Picking a marble out of a bag, replacing it, and picking another marble


  • Rolling a standard 6 sided die and then rolling it again is an example of independent events. Even though the same die is rolled both times, the result of the first roll has no effect on the result of the second roll.
  • Picking two cards out of a standard deck of cards without replacing the first card before drawing the second is an example of dependent events. If you do not replace the first card, then the second card is picked out of 51 cards instead of 52, and it is impossible to draw the same card that was drawn first.
  • Picking two marbles out of a bag with replacement is an example of independent events. The act of replacing the first marble before drawing the second results in the first draw having no effect on the second.

Probability of Greater Than Two Independent Events

The formula for the probability of two independent events occurring P(A and B)=P(A)*P(B) can be extended to more than two independent events - just keep multiplying the individual probabilities. For example, the probability of five independent events occurring is P(A and B and C and D and E) = P(A)*P(B)*P(C)*P(D)*P(E)


Determine the probability of the independent events occurring

  • Rolling an odd number on a standard 6 sided die and picking a red card out of a standard deck of cards and flipping a fair coin and getting heads
  • Picking 5 cards out of a standard deck, with replacement, and getting 5 hearts


  • There are three odd numbers out of the six numbers on the die, so the probability of rolling an odd is 3/6 or 1/2. There are 26 red cards out of 52 cards in a standard deck, so picking a red card has probability 26/52 = 1/2. The probability of flipping a coin and getting heads is also 1/2. So the probability of all three events happening is (1/2)*(1/2)*(1/2) = 1/8.
  • There are 13 hearts out of the 52 cards, so the probability of drawing a heart is 13/52 = 1/4. Replacing the card each time, we get the probability of drawing 5 hearts is (1/4)*(1/4)*(1/4)*(1/4)*(1/4) = 1/1024.

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