Question

Consider a sample space S={E1, E2, E3, E4,} where P(E1) = P(E4) = 0.3 and P(E2) = P(E3) = 0.2

Define the events :

A={E1, E2}

B={E2, E3}

C={E3, E4}

Which of the following pairs of events are independent? Explain.

1. A and B

2. B and C

3. A and C

Answer 1

Two events $A$ and $B$ are called independent if and only if $P(A\cap B)=P(A)P(B)$.

We have:

1. $A\cap B=\{E_2\}$. $P(A\cap B)=0.2$; $P(A)P(B)=0.5\cdot0.4=0.2$. Thus, the events are independent.
2. $C\cap B=\{E_3\}.$ $P(C\cap B)=0.2;$ $P(C)P(B)=0.5\cdot0.4=0.2$ . The events are independent.
3. $A\cap C=\emptyset$ . $P(A\cap C)=0$ ; $P(A)P(C)=0.5\cdot0.5=0.25.$ The events are dependent.