88.
a. The sample space for rolling a die and then tossing a coin consists of 12 outcomes, since there are 6 outcomes for the die and 2 for the coin. It can be represented as:
\[ \{ (1,H), (1,T), (2,H), (2,T), (3,H), (3,T), (4,H), (4,T), (5,H), (5,T), (6,H), (6,T) \} \]
b. \( P(A) \) involves the event of rolling either a three or a four, followed by landing a head. There are 4 outcomes that satisfy this condition: \( (3,H), (4,H) \). Since there are 12 possible outcomes, the probability is:
\[ P(A) = \frac{2}{12} = \frac{1}{6} \]
c. Events \( A \) and \( B \) are not mutually exclusive. Event \( A \) involves rolling a die and then tossing a coin, aiming for a specific number on the die followed by heads on the coin toss. Event \( B \) as described does not relate to the current setup because it involves the first and second tosses landing on heads, which applies to a scenario with multiple coin tosses, not a die roll followed by a coin toss. Therefore, the question seems to have a minor inconsistency with the description of \( B \) as it relates to the setup of a die followed by a coin toss. If \( B \) is intended to be about coin toss outcomes in a different experiment, then its relationship to \( A \) in terms of mutual exclusivity cannot be directly assessed from the information given. If \( B \) were related to coin tosses in this context, it's misdescribed since the experiment involves a die roll followed by a coin toss. Typically, mutual exclusivity between \( A \) and \( B \) in a well-defined scenario means no single outcome can satisfy both \( A \) and \( B \) simultaneously; however, with an unclear definition of \( B \), a precise explanation based on the given information isn't feasible.
89.
a. The sample space for tossing a nickel, a dime, and a quarter, considering H for heads and T for tails, consists of 8 outcomes:
\[ \{ (H,H,H), (H,H,T), (H,T,H), (H,T,T), (T,H,H), (T,H,T), (T,T,H), (T,T,T) \} \]
b. \( P(A) \) involves the event of getting at least two tails. The outcomes that satisfy this condition are: \( (H,T,T), (T,H,T), (T,T,H), (T,T,T) \). Thus, there are 4 outcomes where at least two tails appear, out of 8 possible outcomes. The probability is:
\[ P(A) = \frac{4}{8} = \frac{1}{2} \]
c. Events \( A \) and \( B \) are not mutually exclusive. Event \( A \) requires at least two tails, and event \( B \) specifies the first and second tosses land on heads. An outcome like \( (H,H,T) \) would not satisfy \( A \) since it does not have at least two tails. Therefore, \( A \) and \( B \) can occur independently, but based on their specific definitions, they cannot happen simultaneously. \( A \) demands at least two tails, and \( B \) requires the first two outcomes to be heads, making it impossible for an outcome to satisfy both conditions at the same time. Thus, they are mutually exclusive because no single outcome in the sample space can satisfy both conditions simultaneously, meaning there's no overlap where both \( A \) and \( B \) can occur together.
