A-Level Discrete random variables: worked solution

Question

A bag has 6 red, 6 green, 6 yellow and 6 blue discs (24 in all). Four are drawn without replacement and X counts the different colours obtained, with P(X=2)=465/1771 already known. (b) From this, work out the full probability distribution of X.

Step 1: count the total number of selections. Four discs are chosen from 24 without replacement and the order does not matter, so the number of equally likely selections is C(24,4) = (24 × 23 × 22 × 21)/(4 × 3 × 2 × 1) = 10626. Step 2: find P(X=1), which means all four discs share one colour. Choose which colour is used: 4 ways. Choose 4 of that colour's 6 discs: C(6,4) = 15 ways. Favourable outcomes = 4 × 15 = 60. P(X=1) = 60/10626 = 10/1771, dividing numerator and denominator by 6. Step 3: find P(X=4), which means one disc of every colour. One disc is taken from each colour block and each block holds 6 discs, so favourable outcomes = 6 × 6 × 6 × 6 = 1296. P(X=4) = 1296/10626 = 216/1771, again dividing by 6. Step 4: find P(X=3) from the fact that the probabilities must add to 1. X can only take the values 1, 2, 3, 4, and P(X=2) = 465/1771 is given, so P(X=3) = 1 − 10/1771 − 465/1771 − 216/1771 = (1771 − 10 − 465 − 216)/1771 = 1080/1771. Check: 10 + 465 + 1080 + 216 = 1771, so the four probabilities do sum to 1. The distribution is therefore P(X=1) = 10/1771, P(X=2) = 465/1771, P(X=3) = 1080/1771, P(X=4) = 216/1771.