14a. Two pupils are chosen at random from a group of 4 boys and 5 girls. Find the probability that the two pupils chosen would be boys. Leave your answer in fraction 'a/b.'
b. Twenty percent of the total production of transistors produced by a machine are below standard. If a random sample of six transistors produced by the machine is taken, what is the probability of getting
i. Exactly 2
ii. Exactly 1. Leave your answer in six decimal places " a.bcdefg."
iii. At least 2. Leave your answer in five decimal places " a. bcdeg."
iv. At most 2 standard transistors
14a. Total number of pupils = 4 boys + 5 girls = 9.
Number of ways to choose 2 boys = \(\binom{4}{2} = 6\).
Total number of ways to choose any 2 pupils = \(\binom{9}{2} = 36\).
Probability that both chosen are boys = \(\dfrac{6}{36} = \dfrac{1}{6}\).
14b. This is a binomial distribution with n = 6 trials and p = 0.80 (probability a transistor is standard), q = 0.2 (Prb. of a transistor below standard)
Let Y = number of standard transistors.
P(Y = k) = C(6, k) × (0.8)\(^k\) × (0.2)\(^{(6−k)}\)
i: k = 2 → 15 × 0.64 × 0.0016 = 0.015360
ii: k = 1 → 6 × 0.8 × 0.00032 = 0.001536
iii: P(Y ≥ 2) = 1 − P(Y = 0) − P(Y = 1) = 1 − 0.000064 − 0.001536 = 0.99840
iv: P(Y ≤ 2) = P(Y = 0) + P(Y = 1) + P(Y = 2) = 0.000064 + 0.001536 + 0.015360 = 0.016960
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