Two cards are drawn together at random from a standard pack of 52 playing cards. What is the probability that exactly one card is a spade and the other card is a heart?

Introduction / Context:
This question is about selecting two cards from a full deck and asking for a specific combination of suits: one spade and one heart. It tests knowledge of combinations and understanding that we must count all ways to get one card from each suit, without caring about the order in which the cards are drawn.

Given Data / Assumptions:

• A standard deck has 52 cards.
• There are 13 spades and 13 hearts in the deck.
• Two cards are drawn simultaneously, that is, without order and without replacement.
• Event of interest: one card is a spade and the other is a heart.

Concept / Approach:
The total number of ways to choose 2 cards from 52 is C(52, 2). To have one spade and one heart in the same pair, we must choose one card out of 13 spades and one card out of 13 hearts, so favourable outcomes are 13 * 13. The required probability is the ratio of favourable pairs to total pairs.

Step-by-Step Solution:
Total number of unordered pairs = C(52, 2) = 52 * 51 / 2 = 1326. Number of spades = 13, number of hearts = 13. Favourable ways to choose one spade and one heart = 13 * 13 = 169. Required probability = 169 / 1326. Simplify 169 / 1326 by dividing numerator and denominator by 13: 169 / 1326 = 13 / 102.

Verification / Alternative check:
We can also consider ordered draws: pick any card first and then the second, and then divide by total ordered outcomes. For example, P(spade then heart) = (13/52) * (13/51), and P(heart then spade) = (13/52) * (13/51). Adding gives 2 * 13 * 13 / (52 * 51) which simplifies to 13/102, matching the combination method.

Why Other Options Are Wrong:
3/20, 47/100 and 29/34 do not simplify to 13/102 and arise from incorrect favourable counts or denominators. 29/34 is extremely large and would imply that nearly every pair is one spade and one heart, which is clearly not true.

Common Pitfalls:
Some learners incorrectly treat suits as equally likely for each card and then multiply probabilities for independent suits without adjusting for the second draw. Others forget that the denominator must be based on 52 choose 2, not 52 squared, when working with unordered pairs. Clearly identifying favourable combinations greatly reduces such errors.

Final Answer:
Therefore, the probability that exactly one card is a spade and the other is a heart is 13/102.