Question

An experiment succeeds twice as often as it fails. The probability of success in the next trial is:

• A. \(\frac{1}{3}\)
• B. \(\frac{2}{3}\)
• C. \(1\)
• D. \(0\)

Hint:

If the ratio of success to failure is \(m:n\), then \[ P(\text{Success}) = \frac{m}{m+n}. \] For ratio \(2:1\), \[ P(\text{Success}) = \frac{2}{3}. \]

Answer 1

The Correct Option is B

Concept: Let \[ P(\text{Success})=p \] and \[ P(\text{Failure})=q. \] Since success and failure are complementary events, \[ p+q=1. \]

Step 1: Translate the given condition. The experiment succeeds twice as often as it fails. Therefore, \[ p=2q. \]

Step 2: Use the complementary relation. Substituting \(p=2q\) into \[ p+q=1, \] we obtain \[ 2q+q=1. \] \[ 3q=1. \] \[ q=\frac{1}{3}. \]

Step 3: Find the probability of success. \[ p=2q = 2\left(\frac{1}{3}\right) = \frac{2}{3}. \] Conclusion: \[ \boxed{\frac{2}{3}} \] Hence, the correct answer is Option (B).