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.

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Note
WHAT IS PROBABILITY?

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.

INDEPENDENT EVENTS

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.