Question

A, B, C are aiming to shoot a balloon, A will succeed 4 times out of 5 attempts. The chance of B to shoot the balloon is 3 out of 4 and that of C is 2 out of 3. If the three aim the balloon simultaneously, then find the probability that atleast two of them hit the balloon.

Answer 1

$P\left(A\right)=\frac{4}{5},$ $P\left(B\right)=\frac{3}{4},$ $P\left(C\right)=\frac{2}{3}$

Chance of any two of these hitting the balloon is -

$=P\left(A\right).P\left(B\right).P\left(\overline{C}\right)+P\left(A\right)P\left(\overline{B}\right)P\left(C\right)+P\left(\overline{A}\right).P\left(B\right).P\left(C\right)$

$=\frac{4}{5}\times \frac{3}{4}\times \frac{1}{3}+\frac{4}{5}\times \frac{1}{4}\times \frac{2}{3}+\frac{1}{5}\times \frac{3}{4}\times \frac{2}{3}$

$=\frac{1}{5}+\frac{2}{20}+\frac{1}{10}$

$=\frac{4+2+2}{20}$

$=\frac{8}{20}$

$=\frac{2}{5}.$

Hence, the answer is $\frac{2}{5}.$