$A, B, C$  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.

$A,B,C$  are aiming to shoot a balloon probability of  hitting balloon

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

Probability of at least two of them hit the balloon

$=P(A\cap B\cap \overline{C})+P(A\cap \overline{B}\cap C)+P(\overline{A}\cap B\cap C)+P(A\cap B\cap C)$

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

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

$=\frac{12+8+6+24}{60}=\frac{50}{60}=\frac{5}{6}$