Question

In a tuition batch of 2 students, the probability that \(X\) will pass the examination is \(\frac25\) and that of \(Y\) is \(\frac34\). Assuming independence, what is the probability that neither \(X\) nor \(Y\) will pass?

• A. \(\frac15\)
• B. \(\frac1{10}\)
• C. \(\frac3{20}\)
• D. \(\frac3{10}\)

Hint:

For independent events, multiply probabilities. Always convert “neither” into the probability that each event does not occur.

Answer 1

The Correct Option is C

Concept: For independent events, \[ P(A\cap B)=P(A)\times P(B). \] The probability that neither student passes is the product of their failure probabilities.

Step 1: Find the probability that \(X\) fails.
\[ P(X\text{ fails}) = 1-\frac25 = \frac35. \]

Step 2: Find the probability that \(Y\) fails.
\[ P(Y\text{ fails}) = 1-\frac34 = \frac14. \]

Step 3: Use independence.
\[ P(\text{Neither passes}) = P(X\text{ fails}) \times P(Y\text{ fails}). \] \[ = \frac35\times\frac14. \] \[ = \frac3{20}. \] Hence, \[ \boxed{\frac3{20}}. \]