Question

A problem in Mathematics is given to three students, A, B, C, and their respective probability of solving the problem is $\frac{1}{2},\frac{1}{3}$ and $\frac{1}{4}$. The probability that the problem is solved is

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

Answer 1

The correct option is A

$\frac{3}{4}$

Explanation for the correct option:

Step 1: Find the probability of the first student not solving the question:

Find the complement of the probability of the first student solving the question.

The probability that the first student didn't solve it:

$\begin{array}{rcl}p(A\text{'})& =& 1-\frac{1}{2}\\ & =& \frac{1}{2}\end{array}$

Step 2: Find the probability of the second student not solving the question:

Find the complement of the probability of the second student solving the question.

The probability that the second student didn't solve it:

$\begin{array}{rcl}p(B\text{'})& =& 1-\frac{1}{3}\\ & =& \frac{2}{3}\end{array}$

Step 3: Find the probability of the third student not solving the question:

Find the complement of the probability of the third student solving the question.

The probability that the third student didn't solve it:

$\begin{array}{rcl}p(C\text{'})& =& 1-\frac{1}{4}\\ & =& \frac{3}{4}\end{array}$

Step 4: Find the probability of the none of the students solving the question:

Take the product of the probability of students not solving the question.

So, the probability that none of them solved it all together:

$\begin{array}{rcl}P(S\text{'})& =& \frac{1}{2}·\frac{2}{3}·\frac{3}{4}\\ & =& \frac{1}{4}\end{array}$

Step 5: Find the probability that the problem is solved by at least one of them:

The probability that the problem is solved equals unity subtracted by probability not solved.

$\begin{array}{rcl}P\left(S\right)& =& 1-P(S\text{'})\\ & =& 1-\frac{1}{4}\\ & =& \frac{3}{4}\end{array}$

Hence, option (A) is the correct answer.