Question

A fair coin is tossed 8 times find the probability that it shows heads, exactly 5 times

Answer 1

A fair coin is tossed 8 times find the probability that it shows heads exactly 5 times.

On a given toss (a Bernoulli trial), $p=P\left(head\right)=1/2$ and $q=P\left(tail\right)=1-p=1/2.$

The probability of getting exactly five heads in eight tosses is obtained using the binomial probability formula $P(n-k)$

$=(\frac{n}{k})p^k{q}^{n-k}$

where $n=8,k=5$, and p and q are as given above. So,

P(coin shows exactly S heads in 8 tosses) $=P(n=5)$

$=(\frac{8}{5}){\left(\frac{1}{2}\right)}^5{\left(\frac{1}{2}\right)}^3$

$=56{\left(\frac{1}{2}\right)}^8$

$=56\left(\frac{1}{256}\right)$

$P=\frac{56}{256}=0.21875$