Question

A problem in statistics is given to solve to three students independently. Their chances of solving problem are $\frac{1}{3},\frac{1}{4},\frac{1}{5}$ respectively. Find the probability that the problem is solved.

Answer 1

The random experiments of trying the problem in statistics by the three students are independent.

Let $A,B,C$ be the event of first, second, third student moving problem.
Then, $P\left(A\right)=\frac{1}{3},P\left(B\right)=\frac{1}{4},P\left(C\right)=\frac{1}{5}$
$\therefore$ $P$ (the problem is solved)
$=P\left(AorBorC\right)$
$=1-P\left[\overline{A}\overline{B}\overline{C}\right]$
$=1-P\left(\overline{A}\right)P\left(\overline{B}\right)P\left(\overline{C}\right)$
$=1-\{1-P(A\left)\right\}\{1-P(B\left)\right\}\{1-P(C\left)\right\}$
$=1-(1-\frac{1}{3})(1-\frac{1}{4})(1-\frac{1}{5})$
$=1-\frac{2}{3}\times \frac{3}{4}\times \frac{4}{5}=1-\frac{2}{5}=\frac{3}{5}$