Question

A problem in mathematics is given to $3$ students whose chances of solving individually are $\frac{1}{2},\frac{1}{3}$ and $\frac{1}{4}$. The probability that the problem will be solved at least by one, is.

• A. $\frac{1}{4}$
• B. $\frac{1}{24}$
• C. $\frac{23}{24}$
• D. $\frac{3}{4}$

Answer 1

The correct option is D $\frac{3}{4}$

Let $A,B$ and $C$ be the three students and $P\left(A\right),P\left(B\right),P\left(C\right)$ be their probabilities of solving a problem respectively.

$P\left(A\right)=\frac{1}{2}$, $P\left(B\right)=\frac{1}{3}$, $P\left(C\right)=\frac{1}{4}$

$P[$problem will be solved at least by $1]$

$=$ $1-P\left(\overline{A}\right)P\left(\overline{B}\right)P\left(\overline{C}\right)$

$=$ $1-[1-P(A\left)\right][1-P(B\left)\right][1-P(C\left)\right]$

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

$=$$1-\frac{1}{4}$

$=$$\frac{3}{4}$