AS & A Level Maths

📘 Notes

Independent Events and the Multiplication Law

Independent events do not affect each other.

Two events are independent if the occurrence of one does not change the probability of the other.

If \(A\) and \(B\) are independent, then:

\[ P(A \mid B)=P(A) \qquad \text{and} \qquad P(B \mid A)=P(B) \]

So knowing that one event has happened gives no new information about the other.

For independent events \(A\) and \(B\):

\[ P(A \text{ and } B)=P(A \cap B)=P(A)\times P(B) \]

We multiply the probabilities because the events do not affect each other.

A fair coin is tossed and a fair die is rolled.

Let:

These events are independent.

\[ P(A)=\frac{1}{2}, \qquad P(B)=\frac{1}{6} \]

So:

\[ P(A \cap B)=P(A)\times P(B)=\frac{1}{2}\times\frac{1}{6}=\frac{1}{12} \]

Two events \(A\) and \(B\) are independent if and only if:

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

If this equality is not true, then the events are not independent.