Chapter 15: Q7P (page 734)

(a) Note that (3.4) assumes P(A) is not equal to 0 since $P_{A} (B)$ 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
$P (A) = P_{A} (B)$ 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

Expert verified

(a) $P (A) = P_{B} (A)$ is valid.

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

role="math" localid="1664360840497" $P (E E') = P (E) \times P (E')$

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 $P (A) = P_{B} (A)$

Verifying the given statement

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

Using the conditional probability, $P_{B} (A) = \frac{P (A B)}{P (B)}$ .

From the obtained relations, it is observed that $P (A) = P_{B} (A)$ .

Thus B is independent of A and vice versa.

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

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Short Answer

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Definition of Independent Event

Important Information

Verifying the given statement

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