Question

Two students 𝐴 and 𝐵 are assigned to solve a problem separately. The (conditional) probability that 𝐴 can solve the problem given that 𝐵 cannot solve it, is 1/5. The (conditional) probability that 𝐵 can solve the problem given that 𝐴 can solve the problem is 3/5 . The probability that 𝐴 can solve the problem is 1/10.Then, the probability that 𝐵 can solve the problem is ______ (rounded off to one decimal place).

Answer 1

Correct Answer: 0.8

$$P(A) = \frac{1}{10}, \quad P(B|A) = \frac{3}{5}, \quad P(A|B^c) = \frac{1}{5}$$ Using the law of total probability: $$P(B) = P(B|A) P(A) + P(B|A^c) P(A^c)$$ Since, $$P(A^c) = 1 - P(A) = 1 - \frac{1}{10} = \frac{9}{10}$$ And given: $$P(B|A^c) = 1 - P(A|B^c) = 1 - \frac{1}{5} = \frac{4}{5}$$ Substituting the values: $$P(B) = \frac{3}{5} \times \frac{1}{10} + \frac{4}{5} \times \frac{9}{10}$$ $$= \frac{3}{50} + \frac{36}{50} = \frac{39}{50} = 0.78$$