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What Is a Dependent Event? (Mathematics Lesson)

What Is a Dependent Event?

In probability, two events are dependent if the probability of one event occuring depends on whether the other event occurs.

A Real Example of a Dependent Event

Imagine playing a card trick, where you are asked to pick a card, any card, from a deck of cards.

The probability of picking a Spade can be found. Since 13 out of 52 cards is a Spade, the probability of picking a spade is:

If a Spade is picked but not replaced, what is the probability of picking another Spade?

This time, there are now 12 Spades out of a total of 51 cards. The probability of picking a spade this time is:

The probability of picking a Spade each time depends on whether it has been picked before. It is a dependent event.

Another Real Example of a Dependent Event

Picking colored marbles from a bag, without replacing them, gives another example of dependent events.

The slider below shows how the probabilities of picking a certain colored marble changes depending on whether that color has been picked before.

Visualizing Dependent Events on a Tree Diagram

A tree diagram can be used to illustrate dependent events.

The tree diagram below illustrates someone picking a card from a deck - without replacing the 1st card. At each branch, we are interested in whether the card is Spades (S) or O for Other (for simplicity's sake, we are not interested in whether it as a Club, Diamond or Heart).

The left-most branch shows the result of the 1st pick, and the right-most branches shows the results of the 2nd pick:

The fact that the probabilities aren't the same in the 1st and the two possible 2nd picks shows that each pick is a dependent event. The probabilities depend on previous picks.

Interactive Test

Probability tells us how likely something is to happen. Probability is given as a number between 0 and 1.

A probability of 0 means an event is impossible.

A probability of 1 means an event is certain.

The closer a probability is to 0 the less likely it is. The closer a probability is to 1 the more likely it is.


The opposite to dependent events are independent events.

In this case, each trial does not depend on what happened before.

An example might be tossing a coin. The probability of getting a Head or Tail is ½, regardless of how many times a coin is tossed.

colored marble from a bag without replacing it. Each time a marble is picked out, there will be one less marble in the bag than last time, so there will be a different probability.