Question

A die is thrown $6$ times. If "getting an odd number" is a ''success'', what is the probability of (i) $5$ successes (ii) atmost $5$ successes

Answer 1

Let $X$ be the event of getting an odd number.
Then $X$ has the binomial distribution with $n=6\text{and}p=\frac{1}{2}.$
$\Rightarrow q=1-p=\frac{1}{2}$

$\left(i\right)P\left(5\text{successes}\right)=P(X=5)={}^6{C}_5{\left(\frac{1}{2}\right)}^5\left(\frac{1}{2}\right)=\frac{3}{32}$

$\left(ii\right)P\left(\text{at most}5\text{successes}\right)=P(X\le 5)$
$=1-P(X=6)$
$=1-{}^6{C}_6{\left(\frac{1}{2}\right)}^6$
$=1-\frac{1}{64}=\frac{63}{64}$