Question

(a) Note that (3.4) assumes P(A) is not equal to 0 since ${\mathit{P}}_{\mathbf{A}}\left(B\right)$is meaningless if P(A) = 0.

Assuming both P(A) is not equal to 0 and P(B) is not equal to 0, show that if (3.4) is true, then

$\mathit{P}\mathbf{\left(}\mathit{A}\mathbf{\right)}\mathbf{}\mathbf{=}\mathbf{}{\mathit{P}}_{\mathbf{A}}\mathbf{\left(}\mathit{B}\mathbf{\right)}$that is if B is independent of A, then A is independent of B.

If either P(A) or P(B) is zero, then we use (3.5) to define independence.

(b) When is an event E independent of itself? When is E independent of“not E”?

Short answer

(a) $P\left(A\right)=P_B\left(A\right)$is valid.

(b) E is independent of E when role="math" localid="1664360833595" $P\left(EE\right)=P{\left(E\right)}^2$and is not independent on not E

role="math" localid="1664360840497" $P(EE\text{'})=P\left(E\right)\times P(E\text{'})$

Step-by-step solution

Definition of Independent Event

The events are said to be independent when the occurrence or non-occurrence of any event does not have any effect on the occurrence or non-occurrence of the other event.

When events are independent, apply the formula P(AB) = P(A) . P(B) where A and B are the events.

Important Information

Equation 3.4$P\left(A\right)=P_B\left(A\right)$

Verifying the given statement

When the events A and B are independent of each other then $P\left(AB\right)=P\left(A\right)\times P\left(B\right)$.

Using the conditional probability, $P_B\left(A\right)=\frac{P\left(AB\right)}{P\left(B\right)}$.

From the obtained relations, it is observed that $P\left(A\right)=P_B\left(A\right)$.

Thus B is independent of A and vice versa.

It can be observed that E is independent of E when $P\left(EE\right)=P{\left(E\right)}^2$and is not independent on not E $P(EE\text{'})=P\left(E\right)\times P(E\text{'})$.