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

Concept: For two independent events A and B, the probability that both occur together is the product of their individual probabilities: P(A and B) = P(A) x P(B).

1. The die roll and the coin toss are independent events, since the outcome of one does not affect the outcome of the other.

2. Odd numbers on a standard die are 1, 3, and 5, out of six equally likely faces, so P(die shows an odd number) = 3/6 = 1/2.

3. A coin has two equally likely outcomes, so P(coin shows a head) = 1/2.

4. By the multiplication rule for independent events, P(odd number and head) = (1/2) x (1/2) = 1/4.

Cross-check: Listing the full sample space of (die, coin) pairs gives 6 x 2 = 12 equally likely outcomes. The favourable outcomes are (1, Head), (3, Head), and (5, Head) -- 3 outcomes. So the probability is 3/12 = 1/4, which confirms the result above.