But They Are Two Different Events!

Probability Level 3

One hundred identical coins, each with probability $p$ of showing heads are tossed. If the probability of getting 50 heads equal to that of getting 51 heads, find the value of $p$ .

Clarification : $p$ is a value in the interval $0<p<1$ .

\frac{50}{101}

\frac12

\frac{51}{101}

\frac{49}{101}

\frac{52}{101}

1 solution

Brian Charlesworth
Mar 11, 2016

With the probability of throwing a head being $p$ , the probability of throwing a tail is $1 - p$ . Using the binomial theorem , the given information then translates to the equation

$\dbinom{100}{50} p^{50}(1 - p)^{50} = \dbinom{100}{51}p^{51}(1 - p)^{49} \Longrightarrow \dfrac{100!}{50!*50!}(1 - p) = \dfrac{100!}{51!*49!}p$

$\Longrightarrow \dfrac{51!}{50!}(1 - p) = \dfrac{50!}{49!}p \Longrightarrow 51(1 - p) = 50p \Longrightarrow 51 = 101p \Longrightarrow p = \boxed{\dfrac{51}{101}}$ .

But They Are Two Different Events!

1 solution

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