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Question
State whether the following is True or False :
If X ~ B(n,p) and n = 6 and P(X = 4) = P(X = 2) then p = `(1)/(2)`
Options
True
False
Solution
True
Given, n = 6, P(X = 4) = P(X = 2)
X ~ B(n, p) ≡ X ~ B (6, p)
The p.m.f. of X is given by
p(X = x) = `""^"n""C"_x "p"^x q^("n" - x)`
∴ p(X = x) = `""^6"C"_x "p"^x q^(6 - x), x` = 0, 1, 2, ..., 6
Now, P(X = 4) = P(X = 2)
∴ `""^6"C"_4 "p"^4 "q"^2 = ""^6"C"_2 "p"^2 "q"^4`
∴ `(6!)/(4!2!) "p"^2 = (6!)/(2!4!) "q"^2`
∴ p2 = q2
∴ p = ± q
∴ p =q
∴ p = 1 – p ...[∵ q = 1 – p ]
∴ p + p = 1
∴ 2p = 1
∴ p = `(1)/(2)`.
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