Question

Five dice are thrown simultaneously. If the occurrence of an odd number in a single dice is considered a success, find the probability of maximum three successes.

Answer 1

Let $A$ be the event occuring an odd number and it is success.
$p\left(A\right)=\frac{1}{2},q\left(A\right)=\frac{1}{2}$

$\therefore P($at most $3$ success$)$ or $P($maximum of $3$ success$)$
$=\sum _{r=0}^3$ ${}^n{C}_r{p}^r{q}^{n-r}$ where $n$ is $5$.

${=}^5{C}_0{\left(\begin{array}{c}\frac{1}{2}\end{array}\right)}^5{+}^5{C}_1\left(\begin{array}{c}\frac{1}{2}\end{array}\right){\left(\begin{array}{c}\frac{1}{2}\end{array}\right)}^4{+}^5{C}_2{\left(\begin{array}{c}\frac{1}{2}\end{array}\right)}^2{\left(\begin{array}{c}\frac{1}{2}\end{array}\right)}^3{+}^5{C}_3{\left(\begin{array}{c}\frac{1}{2}\end{array}\right)}^3{\left(\begin{array}{c}\frac{1}{2}\end{array}\right)}^2$
$=\frac{1}{32}+\frac{5}{32}+\frac{10}{32}+\frac{10}{32}$

$=\frac{26}{32}=\frac{13}{16}$.