A-Level Probability: worked solution

Question

A 10-character code has its first 5 characters chosen from the letters {A,B,C,D,E,F,G,H} and its last 5 from the digits {0,1,2,3,4,5,6,7,8,9}, with repeats allowed. (a) How many distinct codes exist? [1] (b) For a code picked at random, find the probability it (i) has exactly three E's and exactly one 5, [2] (ii) begins with E or ends with 4 but not both. [2]

(a) Each of the first 5 characters can be any of the 8 letters, and each of the last 5 can be any of the 10 digits, with repeats allowed, so the numbers of choices multiply: 8 × 8 × 8 × 8 × 8 × 10 × 10 × 10 × 10 × 10 = 8⁵ × 10⁵ 8⁵ = 32768 and 10⁵ = 100000 Number of codes = 32768 × 100000 = 3 276 800 000 (b)(i) All 3 276 800 000 codes are equally likely, so the probability is (favourable codes)/(total codes). Count the favourable ones, treating letters and digits separately. Letters - exactly three E's. First choose WHICH 3 of the 5 letter positions hold the E's: C(5,3) = 10 ways. The remaining 2 positions must NOT be E, so each has 7 choices: 7² = 49. Favourable letter blocks = 10 × 49 = 490 Digits - exactly one 5. Choose which 1 of the 5 digit positions holds the 5: C(5,1) = 5 ways. The remaining 4 positions must not be 5, so each has 9 choices: 9⁴ = 6561. Favourable digit blocks = 5 × 6561 = 32805 The letter block and the digit block are chosen independently, so Favourable codes = 490 × 32805 = 16 074 450 P = 16 074 450 / 3 276 800 000 = 0.0049055... P ≈ 0.00491 (3 s.f.) (b)(ii) Let S be the event "begins with E" and F the event "ends with 4". P(S) = 1/8, since the first character is one of 8 equally likely letters. P(F) = 1/10, since the last character is one of 10 equally likely digits. The first and last positions are filled independently, so P(S and F) = (1/8)(1/10) = 1/80 "S or F but NOT both" is the union with the overlap removed. The union formula P(S) + P(F) - P(S and F) removes the overlap ONCE, so it is still counted once; to exclude it altogether, subtract it a second time: P = P(S) + P(F) - 2P(S and F) P = 1/8 + 1/10 - 2(1/80) = 0.125 + 0.1 - 0.025 P = 0.2