What Is an Independent Event?

In probability, two events are independent if the probability of one event occuring does not depend on whether the other event occurs.

Real Examples of an Independent Event

Imagine tossing a coin. The probability of the coin coming up Heads is:

If I toss the coin a 2nd time, the probability of getting a Heads is:

The probability of getting a Head (or a Tail) is the same each time, it doesn't depend on previous tosses. Each toss is independent of the others.

Similarly, if I roll a pair of dice. The probability that I roll a particular number this time does not depend on the number I rolled last time. Each roll is independent of the others.

Visualizing Independent Events on a Tree Diagram

A tree diagram can be used to illustrate independent events.

The tree diagram below illustrates someone tossing a coin twice, once after the other. At each branch, the coin can either be Heads (H) or Tails (T). The ½ by each branch shows that there is a probability of a ½ each time.

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

The fact that the probabilities are the same in the 1st and the 2nd toss shows that each toss is an independent event.
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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.

DEPENDENT EVENTS

The opposite to independent events are dependent events.

In this case, each trial depends on what happened before.

An example might be picking a 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.