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Chapter 4: Problem 62

Define the following two events for two tosses of a coin: \(A=\) at least one head is obtained \(B=\) both tails are obtained a. Are \(A\) and \(B\) mutually exclusive events? Are they independent? Explain why or why not. b. Are \(A\) and \(B\) complementary events? If yes, first calculate the probability of \(B\) and then calculate the probability of \(A\) using the complementary event rule.

Short Answer

Expert verified
a) Events A and B are mutually exclusive but they are not independent. b) Events A and B are complementary with probability of A = 3/4 and probability of B = 1/4.

Step by step solution

01

Define the Probability Space

Since a coin has two outcomes, head (H) and tail (T), and we are tossing the coin twice, the Sample space (S) = \{HH,HT,TH,TT\}. Each of these outcomes has equal probabilities, so the probability of each is 1/4.
02

Determine if Events A and B are Mutually Exclusive

By definition, two events are mutually exclusive if they cannot happen at the same time. In this case, event A occurs if at least one head is obtained which includes the outcomes {HH, HT, TH} and event B occurs when both tails are obtained which includes the outcome {TT}. They cannot occur at the same time as no outcome belongs to both A and B. Hence, A and B are mutually exclusive.
03

Determine If Events A and B are Independent

Two events are independent if the occurrence of one does not affect the occurrence of the other. In this case, since events A and B are mutually exclusive, they cannot be independent. A cannot occur if B has occurred and vice versa.
04

Determine if Events A and B are Complementary

Two events are complementary if one event occurs if and only if the other does not. In other words, the sum of their probabilities is always 1. The probability of B (\(P(B)\)) is 1/4 (as we have only one favorable event TT in the sample space). The remaining probability must be the probability of A (\(P(A)\)) which includes three outcomes (HH, HT, TH), and hence it is 3/4. Therefore, events A and B are complementary, with \(P(A) = 1 - P(B)\), which satisfies the complementary rule.

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