Five students are to be selected from a large population. If 60% of them are boys and the rest are girls, find the probability that : (a) exactly 3 of them ...
Question 1 Report
Five students are to be selected from a large population. If 60% of them are boys and the rest are girls, find the probability that :
(a) exactly 3 of them are boys;
(b) at least 3 of them are girls.
Answer Details
Selecting \(5\) students, \(P(\text{boy}) = 0.6\) and \(P(\text{girl}) = 0.4\). Use the binomial distribution.
(a) Exactly 3 boys (so 2 girls):
\[\binom{5}{3}(0.6)^3(0.4)^2 = 10(0.216)(0.16) = 0.3456 \approx 0.346.\]
(b) At least 3 girls means \(3\), \(4\) or \(5\) girls, with \(P(\text{girl}) = 0.4\):
\[P(3) = \binom{5}{3}(0.4)^3(0.6)^2 = 10(0.064)(0.36) = 0.2304,\]
\[P(4) = \binom{5}{4}(0.4)^4(0.6) = 5(0.0256)(0.6) = 0.0768,\]
\[P(5) = (0.4)^5 = 0.01024.\]
Sum: \(0.2304 + 0.0768 + 0.01024 = 0.31744 \approx 0.317\).
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