A-Level Probability: worked solution

Question

Using the wardrobes (A: T1, T2 formal, T3, T4 casual; B: P1, P2 formal, P3 casual), Without replacement, Nancy selects three outfits one after another. Find the probability that exactly two of them pair a formal blouse with formal pants.

Step 1 - list what is available. Blouses (wardrobe A): T1 and T2 are formal, T3 and T4 are casual, so 4 blouses of which 2 are formal. Pants (wardrobe B): P1 and P2 are formal, P3 is casual, so 3 pairs of pants of which 2 are formal. An outfit is one blouse worn with one pair of pants, and nothing is returned, so the three outfits use 3 of the 4 blouses and all 3 pairs of pants. Step 2 - see what 'exactly two formal outfits' forces. A formal outfit needs a formal blouse AND formal pants. There are only 2 formal blouses and only 2 formal pairs of pants, so two formal outfits use up both of each. The outfit that is left must then take the casual pants P3, so it CANNOT be formal. So once two formal outfits have happened, 'exactly two' is automatic - there is no separate case to exclude. Step 3 - probability along one fixed order, formal then formal then the odd one out. First outfit formal: 2 of the 4 blouses are formal and 2 of the 3 pairs of pants are formal, and the blouse and the pants are chosen independently, so P = (2/4)(2/3) = 1/3. Second outfit formal, given the first was: one formal blouse has gone, leaving 1 formal out of 3 blouses, so 1/3; one formal pair has gone, leaving 1 formal out of 2 pairs, so 1/2. P = (1/3)(1/2) = 1/6. Third outfit: only the casual pants P3 remain, so it is certainly not formal. P = 1. Probability for this order = (2/4)(2/3) × (1/3)(1/2) × 1 = (1/3)(1/6) = 1/18. Step 4 - allow for the order in which the non-formal outfit appears. The non-formal outfit could be the 1st, 2nd or 3rd selection. That is the number of distinct arrangements of the list (F, F, N), which is 3!/2! = 3. (Each order carries the same probability - check F, N, F: (1/3) × (2/3)(1/2) × (1/2)(1) = 1/18.) Step 5 - multiply. P(exactly two formal outfits) = 3 × 1/18 = 1/6.