From a deck of 52 cards, two cards are randomly chosen. What is the…

Concept

The number of ways to choose 2 objects from a set of n (order irrelevant) is the combination nC2 = n(n-1)/2. When a single object can satisfy two different required properties at once (here, the Queen of Diamonds is both "a queen" and "a diamond"), the favourable count must be split into mutually exclusive cases so that no pair is counted twice, then the cases are added (the addition rule for disjoint events).

Application

We need, among the two cards chosen, one card that is a queen and the other that is a diamond.

1. Total ways to choose any 2 cards from 52: 52C2 = (52 * 51) / 2 = 1326.

2. Case 1 - the queen card IS the Queen of Diamonds: choose it (1 way) and pair it with any of the other 12 diamonds (the Queen of Diamonds itself is excluded from this second pick): 1 * 12 = 12 ways.

3. Case 2 - the queen card is one of the other three queens (Hearts, Spades, Clubs): choose one of them (3 ways) and pair it with any of the 13 diamonds: 3 * 13 = 39 ways.

4. These two cases cannot overlap (a pair falls in exactly one of them), so total favourable pairs = 12 + 39 = 51.

5. Probability = 51 / 1326 = 1 / 26 (divide numerator and denominator by 51).

Cross-check

Partition the 52 cards into four disjoint groups: the Queen of Diamonds (1 card), the other three queens (3 cards), the other twelve diamonds (12 cards), and the remaining thirty-six cards (neither a queen nor a diamond). Count every unordered pair across groups that pairs a "queen" card with a "diamond" card: {Queen of Diamonds + another queen} = 1 x 3 = 3 pairs (Queen of Diamonds used in its diamond role), {Queen of Diamonds + another diamond} = 1 x 12 = 12 pairs (Queen of Diamonds used in its queen role), {another queen + another diamond} = 3 x 12 = 36 pairs. Total = 3 + 12 + 36 = 51, matching the case-based count above and confirming 51 / 1326 = 1 / 26.