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13. probability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths13. probability1 Mark
MCQ
Suppose $X$ follows a binomial distribution with parameters $n$ and $p$, where $0 < p < 1.$ If $\frac{{P\,(X = r)}}{{P\,(X = n - r)}}$ is independent of $n$ and $r$, then
  • ✓
    $p = \frac{1}{2}$
  • B
    $p = \frac{1}{3}$
  • C
    $p = \frac{1}{4}$
  • D
    None of these
✓

Answer

Correct option: A.
$p = \frac{1}{2}$
a
(a) We have,
$\frac{{P(X = r)}}{{P(X = n - r)}} = \frac{{{}^n{C_r}{p^r}{{(1 - p)}^{n - r}}}}{{{}^n{C_{n - r}}{p^{n - r}}{{(1 - p)}^r}}} = {\left( {\frac{1}{p} - 1} \right)^{n - 2r}}$
Note that $\frac{1}{p} - 1 > 0.$ Therefore the ratio will be independent of $n$ and $r,$ if $\frac{1}{p} - 1 = 1$ or $p = \frac{1}{2}.$

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