Question

Three independent events, $A_1,A_2$ and $A_3$ occur with probabilities $P\left(A_i\right)=\frac{1}{1+i},i=1,2,3$. What is the probability that at least one of the three events occurs?

• A. $\frac{1}{4}$
• B. $\frac{2}{3}$
• C. $\frac{3}{4}$
• D. $\frac{1}{24}$

Answer 1

Answer: (C)

$\frac{3}{4}$

$P\left(A_i\right)=\frac{1}{1+i}$

$\therefore P\left(A_1\right)=\frac{1}{1+1}=\frac{1}{2},P\left(A_2\right)=\frac{1}{3},P\left(A_3\right)=\frac{1}{4}$

Probability that at least one event occurs = 1 - Probability that none of $A_1,A_2$ or $A_3$ occur

$=1-P(\overline{A_1}\cap \overline{A_2}\cap \overline{A_3})$

$=1-P\left(\overline{A_1}\right)P\left(\overline{A_2}\right)P\left(\overline{A_3}\right)$ ...($A_1,A_2$ and $A_3$ are independent events)

$=1-(1-\frac{1}{2})(1-\frac{1}{3})(1-\frac{1}{4})$

$=1-(\frac{1}{2}\times \frac{2}{3}\times \frac{3}{4})$

$=\frac{3}{4}$

Answer is option (C)