Topic Review — Probability — Solutions

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1. Single event probability

   1. Marble bag (4R, 6B, 2W; total 12): (i) P(red) = 4/12 = 1/3   (ii) P(blue) = 6/12 = 1/2   (iii) P(white) = 2/12 = 1/6   (iv) P(not blue) = 6/12 = 1/2
   2. Die rolled: (i) P(odd) = 3/6 = 1/2   (ii) P(>2) = 4/6 = 2/3   (iii) P(not 4) = 5/6
   3. Card drawn: (i) P(ace) = 4/52 = 1/13   (ii) P(red face card) = 6/52 = 3/26   (iii) P(not heart) = 39/52 = 3/4
   4. Spinner 1–8: (i) Primes: 2,3,5,7 → P = 4/8 = 1/2   (ii) Factors of 8: 1,2,4,8 → P = 4/8 = 1/2   (iii) >6: 7,8 → P = 2/8 = 1/4

2. Complementary events

   1. P(A) = 0.42: P(A′) = 1 − 0.42 = 0.58
   2. P(B′) = 7/12: P(B) = 1 − 7/12 = 5/12
   3. P(not tails): P(not tails) = 1 − 1/2 = 1/2
   4. P(not raining): P(not rain) = 100% − 35% = 65%
   5. P(C) with A=0.15, B=0.45: P(C) = 1 − 0.15 − 0.45 = 0.40
   6. Non-defective in 400: P(non-defective) = 1 − 0.03 = 0.97; Expected = 0.97 × 400 = 388

3. Tree diagrams and two-step experiments

   1. Coin then die: 2 coin outcomes × 6 die outcomes = 12 outcomes total
   2. Spinner R/B spun twice: (i) P(RR) = 1/2 × 1/2 = 1/4   (ii) P(at least one B) = 1 − P(RR) = 3/4
   3. 3G 2O, without replacement: (i) P(GG) = 3/5 × 2/4 = 6/20 = 3/10   (ii) P(one each) = P(GO)+P(OG) = 3/5×2/4 + 2/5×3/4 = 6/20+6/20 = 12/20 = 3/5
   4. Two dice: (i) P(3,3) = 1/6×1/6 = 1/36   (ii) Sum=7: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) = 6/36 = 1/6   (iii) Sum<4: (1,1),(1,2),(2,1) = 3/36 = 1/12

4. Experimental probability and relative frequency

   1. Coin: 44 heads in 80: P(heads) = 44/80 = 11/20 = 0.55; theoretical = 0.5; experimental is close but higher
   2. Spinner: 18 red in 50: (i) Relative freq = 18/50 = 9/25 = 0.36   (ii) Expected in 200 = 0.36 × 200 = 72
   3. Expected rolls < 3 in 120: P(<3) = 2/6 = 1/3; Expected = 1/3 × 120 = 40
   4. 15 faulty in 300: P(faulty) = 15/300 = 1/20 = 0.05; In 2000: 0.05 × 2000 = 100 faulty

5. Mixed problem solving

   1. Activities {Art, Drama, Science} × Snacks {Apple, Banana}:
      1. 3 × 2 = 6 combinations
      2. P(Drama and Banana) = 1/6
      3. P(not Art): Art combinations = 2 out of 6; P(not Art) = 4/6 = 2/3
   2. 5R 3B without replacement:
      1. P(same): P(RR)+P(BB) = (5/8×4/7)+(3/8×2/7) = 20/56+6/56 = 26/56 = 13/28
      2. P(at least one R) = 1−P(BB) = 1−6/56 = 50/56 = 25/28
   3. P(score) = 0.72, 25 attempts: Expected scores = 0.72 × 25 = 18; Expected misses = 25 − 18 = 7
   4. Theoretical vs experimental: Theoretical probability is calculated from equally likely outcomes using maths (e.g. P(heads) = 1/2 for a fair coin). Experimental probability is found by actually conducting trials and recording results. Example where only experimental applies: estimating P(a drawing pin lands point up) — there is no way to calculate this theoretically, so we must experiment.