A die is rolled and a coin is tossed. Find the probability that the die shows…

When two events are joined by 'and' — both must happen together — and the events come from independent experiments (rolling a die, tossing a coin), the compound probability equals the product of each event's individual probability: P(A and B) = P(A) × P(B). This also matches directly counting favourable outcomes in the combined sample space and dividing by the total number of outcomes.

1. The sample space S lists every outcome of rolling a die (6 faces) together with tossing a coin (2 faces): S = {(1,H),(2,H),(3,H),(4,H),(5,H),(6,H),(1,T),(2,T),(3,T),(4,T),(5,T),(6,T)}, so the total number of outcomes is n(S) = 6 × 2 = 12.

2. Event E is 'die shows an odd number AND coin shows a head'. The odd faces are 1, 3, 5, so the favourable outcomes are (1,H), (3,H), (5,H), giving n(E) = 3.

3. P(E) = n(E)/n(S) = 3/12 = 1/4.

Cross-check with the multiplication rule: P(odd number) = 3/6 = 1/2 and P(head) = 1/2; since the die roll and coin toss are independent events, P(odd and head) = 1/2 × 1/2 = 1/4 — the same result as the direct count.