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Question
Out of 750 families with 4 children each, how many families would be expected to have atmost 2 girls
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Solution
Assume equal probabilities for boys and girls
Let p be the probability of having a boy
Let x be the random variable for getting either a boy or a girl
∴ p = `1/2` and q = `1/2` and n = 4
In binomial distribution P(X = 4) = ncxpxqn-x
Here the binomial distribution is P(X= x) = `4"C"_x (1/2)^x (1/2)^("n" - x)`
P(almost 2 girls) = P(X ≤ 2)
= P(X = 0) = P(X = 1) + P(X = 2)
= `4"C"_0 (1/2)^0(1/2)^(4 - 0) + 4"C"_1 (1/2)^1 (1/2)^(4 - 1) + 4"C"_2 (1/2)^2 (1/4)^(4 - 2)`
= `(1)(1)(1/2)^4 + 4(1/2)(1/2)^3 + 6(1/2)^2(1/2)^2`
= `(1/16) + 4(1/16) + 6(1/16)`
= `1/16 [1 + 4 + 6]`
= `11/16`
= 0.6875
For 750 families P(x ≤ 2) = 0.6875 × 750
= 515.625
= 516 .......(approximately)
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