Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 12
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# If for a Binomial Distribution P (X = 1) = P (X = 2) = α, Write P (X = 4) in Terms of α. - Mathematics

#### Question

If for a binomial distribution P (X = 1) = P (X = 2) = α, write P (X = 4) in terms of α.

#### Solution

$\text{ For binomial distribution of X } ,$
$P(X = r) = ^{n}{}{C}_r (p )^r (q )^{n - r} , r = 0, 1, 2, . . . , n$
$P(X = 1) = np(q )^{n - 1}$
$P(X = 2) =^{n}{}{C}_2 p^2 (q )^{n - 2}$
$\Rightarrow np(q )^{n - 1} = ^{n}{}{C}_2 p^2 (q )^{n - 2} = \alpha$
$\text{ Simplifying the above equation we get,}$
$q = \frac{n - 1}{2}p$
$\Rightarrow 2q = np - p$
$\text{ On putting, q = 1 - p we get }$
$2 - 2p = np - p$
$p(n + 1) = 2 . . . . . (i)$
$\text{ Also} , P(X = 1) = \alpha$
$\Rightarrow np(1 - p )^{n - 1} = \alpha . . . . . (ii)$

Note: We cannot find the value of n as (i) and (ii) are not linear and hence we cannot find the value of P(X = 4)

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