Question

south african captain lost th etoss of a coin 13 times out of 14 . the chance of this happening is?

Answer 1

Task. South african captain lost the toss of a coin 13 times out of 14. Find the chance of this happening is?

Proof. Let $X$ be the random variable equal to the number of wins the toss of a coin out of 14 tosses. Assuming that the chances to loss and to win are equal, so their probabilities are equal to 0.5, we obtain that $X$ ras binomial distribution with $n=14$ $p=0.5$, and $q=1-p-0.5$. Hence the chance to lost $k$ times out of $n$ is equal to

$P(X=k)=C_{n}^{k}p^{n-k}q^{k},$

where

$C_{n}^{k}=\frac{n!}{k!(n-k)!}$

is the binominal coefficient, and $k!=k(k-1)\cdots 2\cdot 1$.

For $k=13$ we obtain that

$P(X=13)=C_{14}^{13}\cdot 0.5^{1}\cdot 0.5^{13}=\frac{14!}{13!\cdot 1!}\cdot 0.5^{14}=\frac{14\cdot 13\cdots 2\cdot 1}{13\cdots 2\cdot 1\cdot 1}\cdot 0.5^{14}=14\cdot 0.5^{14}\approx 0.00085449.$