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

Step 1: Understand the phrase neither passes.
The event neither $X$ nor $Y$ passes means $X$ fails AND $Y$ fails at the same time. So we need the probability that both fail.
Step 2: Note the passing probabilities.
We are given $P(X\text{ passes})=\frac25$ and $P(Y\text{ passes})=\frac34$.
Step 3: Convert to failing probabilities.
Failing is the complement of passing, so $P(X\text{ fails})=1-\frac25=\frac35$ and $P(Y\text{ fails})=1-\frac34=\frac14$.
Step 4: Use independence.
The two students are independent, so the chance both fail is the product of the individual failure chances.
Step 5: Multiply the failure probabilities.
\[ P(\text{neither passes})=\frac35\times\frac14=\frac{3}{20}. \]
Step 6: Sanity check the size of the answer.
Since $X$ is more likely to fail but $Y$ is quite likely to pass, a small value like $\frac{3}{20}$ is reasonable.
\[ \boxed{\frac{3}{20}} \]