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Home  >>  JEEMAIN and AIPMT  >>  Mathematics  >>  Class12  >>  Probability
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If $X$ follows a binomial distribution with parameters $n$ and $p$. $p \lt 0 \lt 1$ If $ P(X=r)/P(X=n-r)$ is independent of $ n \:$ and$\: r$ , then value of p is :

$\begin {array} {1 1} (A)\;\large\frac{1}{2} & \quad (B)\;\large\frac{1}{3} \\ (C)\;\large\frac{1}{4} & \quad (D)\;None\: of \: these \end {array}$

 

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1 Answer

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$ \large\frac{P(x=r)}{P(X=n-r)} = \large\frac{nC_r.p^r.(1-p)^{n-r}}{nC_{n-r}.p^{n-r}.(1-p)^r}$
$ = \large\frac{(1-p)^{n-2r}}{p^{n-2r}}$
$ = \bigg( \large\frac{1}{p}-1 \bigg)^{n-2r}$ But since $p <1, \: \large\frac{1}{p}$$-1 >0$
$ \therefore $ The ratio is independent of $ n $ and $r$ if $ \large\frac{1}{p}$$-1=1$
or if $p= \large\frac{1}{2}$
Ans : (A)
answered Jan 6, 2014 by thanvigandhi_1
edited Mar 26, 2014 by rvidyagovindarajan_1
 

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