A-Level Discrete random variables: worked solution

Question

For the wardrobe scoring (sum if formalities match, product if not), give the probability distribution of X.

Method: X is decided entirely by which pair of garments is chosen, so list every equally likely pair, apply the scoring rule to each one, then add up the probabilities of the pairs that give the same score. Step 1: Count the outcomes. Every possible pair of garments is equally likely and there are 12 of them, so each single pair has probability 1/12. Every probability in the final distribution must therefore be a whole number of twelfths. Step 2: Score each pair, one at a time, by checking the two formalities. - If the two formalities are the SAME, the score is the SUM of the two numbers. - If the two formalities are DIFFERENT, the score is the PRODUCT of the two numbers. Write the resulting score next to each of the 12 pairs. Step 3: Collect the pairs that give equal scores. The distinct scores that occur are 2, 3, 4, 6, 7 and 8, and the tally of the 12 pairs is score 2: 1 pair score 3: 4 pairs score 4: 2 pairs score 6: 3 pairs score 7: 1 pair score 8: 1 pair Step 4: Turn each tally into a probability by dividing by 12. P(X = 2) = 1/12 P(X = 3) = 4/12 = 1/3 P(X = 4) = 2/12 = 1/6 P(X = 6) = 3/12 = 1/4 P(X = 7) = 1/12 P(X = 8) = 1/12 Step 5: Check the total, which is how you catch a missed or double-counted pair. 1/12 + 4/12 + 2/12 + 3/12 + 1/12 + 1/12 = 12/12 = 1. Probability distribution of X: x : 2 3 4 6 7 8 P(X = x): 1/12 1/3 1/6 1/4 1/12 1/12