Question

P, Q, and R try to hit the same target one after another. Their probabilities of hitting are \( \frac{2}{3}, \frac{3}{5}, \frac{5}{7} \) respectively. Find the probability that the target is hit by P or Q but not by R.

• A. \( \frac{26}{105} \)
• B. \( \frac{79}{105} \)
• C. \( 0 \)
• D. \( \frac{75}{105} \)

Hint:

For probability problems involving multiple independent events, use union and intersection formulas.

Answer 1

The Correct Option is A

Step 1: Probability of P or Q hitting the target
\[ P(A) = P(\text{P hits}) + P(\text{Q hits}) - P(\text{P and Q hit}) \] \[ = \frac{2}{3} + \frac{3}{5} - \left(\frac{2}{3} \times \frac{3}{5} \right) \] \[ = \frac{10}{15} + \frac{9}{15} - \frac{6}{15} = \frac{13}{15} \] Step 2: Probability of R not hitting the target
\[ P(R' ) = 1 - P(R) = 1 - \frac{5}{7} = \frac{2}{7} \] Step 3: Required Probability
\[ P(A) \times P(R') = \frac{13}{15} \times \frac{2}{7} = \frac{26}{105} \] Thus, the correct answer is \( \frac{26}{105} \).