AS & A Level Maths

📘 Notes

Dependent Events and Conditional Probability

Dependent events are events where the occurrence of one affects the probability of the other.

1. What are dependent events?

Two events are dependent if one event changes the probability of the other.

A common example is selection without replacement. After the first item is chosen, the numbers in the sample space change, so the probability of the second choice also changes.

\[ \text{For dependent events, } P(B\mid A)\ne P(B) \]

2. Conditional probability

Conditional probability is the probability of an event given that another event has already happened.

The probability of \(B\) given \(A\) is written as:

\[ P(B\mid A)=\frac{P(A\cap B)}{P(A)} \]

This formula is used when \(P(A)\ne 0\).

3. The multiplication law of probability

The multiplication law is used to find the probability that “this and that” happens.

For any two events \(A\) and \(B\):

\[ P(A\cap B)=P(A)\times P(B\mid A) \]

It can also be written as:

\[ P(A\cap B)=P(B)\times P(A\mid B) \]

The multiplication law for independent events is a special case of this, because then \(P(B\mid A)=P(B)\).

4. Example: selection without replacement

A bag contains 3 red balls and 2 blue balls. Two balls are chosen without replacement.

Let:

\(A\): the first ball is red
\(B\): the second ball is red

First:

\[ P(A)=\frac{3}{5} \]

If the first ball is red, then 2 red balls remain out of 4 balls:

\[ P(B\mid A)=\frac{2}{4}=\frac{1}{2} \]

So:

\[ P(A\cap B)=P(A)\times P(B\mid A)=\frac{3}{5}\times\frac{1}{2}=\frac{3}{10} \]

The events are dependent because the first selection changes the probability of the second selection.

5. Exam tips