Question

Let \( A \) and \( B \) be events with \( P(A)=1 / 3 \) and \( P(B)=2 / 3 \). Let \( N \) be the number of events from among \( A \) and \( B \) that occur. (For instance, if \( A \) occurs but \( B \) does not, then \( N=1 \).) Suppose that the occurrence of events in \( A \) and \( B \) are independent. Find, for example, the probability that no events occur.

Answer 1

( 2/9 )

Explanation:

Okay, so since the events A and B are independent, the probability that both of them do not occur is the product of their individual probabilities of not occurring.

The probability of event A not occurring, denoted as ( P(A') ), is equal to 1 minus the probability of A occurring. Similarly, the probability of event B not occurring, denoted as ( P(B') ), is equal to 1 minus the probability of B occurring.

So, ( P(A') = 1 - P(A) = 1 - 1/3 = 2/3 ) and ( P(B') = 1 - P(B) = 1 - 2/3 = 1/3 ).

Since A and B are independent, the probability that both A and B do not occur is simply the product of their individual probabilities of not occurring:

[ P(N = 0) = P(A' cap B') = P(A')P(B') = (2/3) times (1/3) = 2/9 ]

So, the probability that no events occur is ( 2/9 ).

Hope this helps! :)