Independent Events

If two events A and B are independent of each other, then the probability that they both occur is given by

That is, the probability of them both happening is found by multiplying together their individual probabilities.

Example

What is the probability of picking a spade from a normal pack of cards and rolling an odd number on a die?

So the probability of both occurring is

Dependent Events

When two events are dependent on one another, it becomes more complicated to find the probability of them both occurring. This is because, as soon as event A occurs, the probability that B occurs changes.

To use the example that introduced dependent events. The probability of being late for school, over the entire year, P(L), might be 0.01. However, as soon as it snows, the chance of you being late immediately increases, to say 0.05. It is this probability that you would need to use in the calculation.

This second probability is called a conditional probability.

We write this

The formula that we now use to find the probability of both it snowing and being late, is

Equally we could use

Example

If a menber of the sixth form is chosen at random, the probability that they study Maths is 0.1.

The probability that they study Art is 0.08, but the probability that a Maths student also studies Art is only 0.03.

Find the probability that a student chosen at random will study both Maths and Art, and also the probability that an Art student will also study Maths.

First write down the information that we have in symbols

To answer the first question

So the probability that a student chosen at random will study both Maths and Art is 0.003.

We use this result to answer the second part of the question.

So the probability that an Art student will also study Maths is 0.0375

On the next page we will continue to look at combining events and finding the probability that either A or B (or both) occur.

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