Question

The probability that A, B and C can solve a problem are 4/5, 2/3 and 3/7 respectively. The probability of problem being solved by A and B is 8/15, B and C is 2/7, A and C is 12/15. The probability of problem being solved by all three is 8/35. Find the probability that problem is not solved by anyone

Answer 1

Answer on Question #53690 – Math – Statistics and Probability

The probability that A, B and C can solve a problem are 4/5, 2/3 and 3/7 respectively. The probability of problem being solved by A and B is 8/15, B and C is 2/7, A and C is 12/15. The probability of problem being solved by all three is 8/35. Find the probability that problem is not solved by anyone

Given:

$$p(A) = \frac{4}{5} \qquad p(B) = \frac{2}{3} \qquad p(C) = \frac{3}{7}$$ $$p(A \cap B) = \frac{8}{15} \qquad p(B \cap C) = \frac{2}{7} \qquad p(A \cap C) = \frac{12}{15}$$ $$p(A \cap B \cap C) = \frac{8}{35}$$

**Find:** $p(A \cup B \cup C)$

Solution

$$p(A \cup B \cup C) = p(A) + p(B) + p(C) - p(A \cap B) - p(A \cap C) - p(B \cap C) + p(A \cap B \cap C) = \frac{4}{5} + \frac{2}{3} + \frac{3}{7} - \frac{8}{15} - \frac{2}{7} - \frac{12}{15} + \frac{8}{35} = \frac{2}{15} + \frac{1}{7} + \frac{8}{35} = \frac{2}{15} + \frac{13}{35} = \frac{53}{105}.$$ $$p(\overline{A \cup B \cup C}) = 1 - p(A \cup B \cup C) = 1 - \frac{53}{105} = \frac{52}{105}.$$

**Answer:** $\frac{52}{105}$

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