(a) The probability that a man wins a race is 0.8. In four different races, what is the probability that he wins : (i) all races ; (ii) no race ; (iii) at m...
Question 1 Report
(a) The probability that a man wins a race is 0.8. In four different races, what is the probability that he wins : (i) all races ; (ii) no race ; (iii) at most 3 races ?
(b) A class consists of 5 girls and 10 boys. If a committee of 5 is chosen at random from the class, find the probability that :
(i) 3 boys are selected ; (ii) at least one girl is selected.
Answer Details
(a) Winning is binomial with \( p = 0.8,\ q = 0.2,\ n = 4 \).
(i) Wins all races: \( p^4 = 0.8^4 = 0.4096. \)
(ii) Wins no race: \( q^4 = 0.2^4 = 0.0016. \)
(iii) Wins at most 3 races \( = 1 - P(\text{all 4}) = 1 - 0.4096 = 0.5904. \)
(b) Committee of 5 chosen from 5 girls and 10 boys (15 people). Total \( = \binom{15}{5} = 3003 \).
(i) Exactly 3 boys (so 2 girls): \[ \frac{\binom{10}{3}\binom{5}{2}}{3003} = \frac{120 \times 10}{3003} = \frac{1200}{3003} = \frac{400}{1001} \approx 0.400. \]
(ii) At least one girl \( = 1 - P(\text{no girl}) = 1 - \dfrac{\binom{10}{5}}{3003} = 1 - \dfrac{252}{3003} = \dfrac{2751}{3003} = \dfrac{917}{1001} \approx 0.916. \)
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