First slide

Random variable and probability distribution

Question

One hundred identical coins, each with probability $p$ , of showing up heads are tossed once. If 0 < $p$ < 1 and the probability of heads showing on 50 coins is equal to that of heads showing on 51 coins, then value of $p$ is

Moderate

A

$\frac{1}{2}$
B

$\frac{49}{101}$
C

$\frac{50}{101}$
D

$\frac{51}{101}$

Solution

$\begin{matrix} P (x = 50) = P (x = 51) \\ \Rightarrow^{100} C_{50} (p)^{50} (1 - p)^{50} =^{100} C_{51} (p)^{51} (1 - p)^{49} [f o r m u l a P (x = r) = c (n, r) q^{n - r} \cdot p^{r}] \\ \Rightarrow \frac{1 - p}{p} = \frac{50}{51} (\sin c e \frac{C (n, r)}{C (n, r - 1)} = \frac{n - r + 1}{r}) \\ \Rightarrow p = \frac{51}{101} \end{matrix}$

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