An experiment consists of first rolling a die and then tossing a coin: a. How many elements are there in the sample space? b. Let A be the event that either a \( 1,2,3 \) or 4 is rolled first, followed by landing a tail on the coin toss. \( P(A)=\square \) Present c. Let \( B \) be the event that an even number is rolled, followed by landing a tail on the coin toss. Are the events A and B mutually exclusive? No, they are not Mutually Exclusive Yes, they are Mutually Exclusive Hint: Mutually Exclusive

**a. Sample Space Size** - The die has \(6\) possible outcomes. - The coin has \(2\) possible outcomes. - Thus, the number of elements in the sample space is \[ 6 \times 2 = 12. \] **b. Probability of Event \(A\)** - Event \(A\) is defined as rolling a \(1, 2, 3,\) or \(4\), followed by landing tails. - The probability of rolling one of these \(4\) outcomes is \[ \frac{4}{6} = \frac{2}{3}. \] - The probability of obtaining tails on the coin toss is \[ \frac{1}{2}. \] - Since these events are independent, the probability \(P(A)\) is \[ \frac{2}{3} \times \frac{1}{2} = \frac{2}{6} = \frac{1}{3}. \] **c. Mutual Exclusivity of Events \(A\) and \(B\)** - Event \(A\): Rolling a \(1, 2, 3,\) or \(4\) and tossing tails. - Event \(B\): Rolling an even number (i.e., \(2, 4,\) or \(6\)) and tossing tails. To check whether \(A\) and \(B\) are mutually exclusive, we look for any common outcomes. - The outcomes in \(A\) with tails are: \[ (1,\text{T}),\ (2,\text{T}),\ (3,\text{T}),\ (4,\text{T}). \] - The outcomes in \(B\) with tails are: \[ (2,\text{T}),\ (4,\text{T}),\ (6,\text{T}). \] - The common outcomes between \(A\) and \(B\) are: \[ (2,\text{T}) \text{ and } (4,\text{T}). \] Since there are common outcomes (i.e., the intersection is not empty), the events \(A\) and \(B\) are **not mutually exclusive**.