Question

If \(n(A) = 2\), \(P(A) = \frac{1}{5}\), then find \(n(S) = ?\)

• A. 10
• B. \(\dfrac{5}{2}\)
• C. \(\dfrac{2}{5}\)
• D. \(\dfrac{1}{3}\)

Hint:

Always use \(P(A) = \frac{n(A)}{n(S)}\) for basic probability problems — rearrange it as \(n(S) = \frac{n(A)}{P(A)}\) when total outcomes are required.

Answer 1

The Correct Option is A

Step 1: Recall the formula for probability.
The probability of an event \(A\) is given by: \[ P(A) = \frac{n(A)}{n(S)} \] where \(n(A)\) is the number of favorable outcomes and \(n(S)\) is the total number of outcomes in the sample space.
Step 2: Substitute the given values.
We are given \(n(A) = 2\) and \(P(A) = \frac{1}{5}\). \[ \frac{1}{5} = \frac{2}{n(S)} \] Step 3: Solve for \(n(S)\).
Multiply both sides by \(n(S)\): \[ n(S) \times \frac{1}{5} = 2 \] \[ n(S) = 2 \times 5 = 10 \] Step 4: Conclusion.
Therefore, the total number of outcomes in the sample space is \(n(S) = 10.\)
Final Answer: \[ \boxed{n(S) = 10} \]