Two friends A and B meet every weekend either | vedclass

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Question

(A) Let $E_1$ be the event that $A$ and $B$ meet at a party, $E_2$ be the event that $A$ and $B$ meet at a sports club, and $D$ be the event that $A$

Vedclass · Accepted Answer

$\frac{86}{135}$

Vedclass · Answer

(A) Let $E_1$ be the event that $A$ and $B$ meet at a party, $E_2$ be the event that $A$ and $B$ meet at a sports club, and $D$ be the event that $A$ and $B$ dine together. Given $P(E_2) = \frac{4}{9}$, so $P(E_1) = 1 - \frac{4}{9} = \frac{5}{9}$. The conditional probabilities are $P(D|E_2) = \frac{2}{5}$ and $P(D|E_1) = \frac{1}{3}$. Using the law of total probability, the probability that they dine together is: $P(D) = P(E_1) \cdot P(D|E_1) + P(E_2) \cdot P(D|E_2)$ $P(D) = \frac{5}{9} 	imes \frac{1}{3} + \frac{4}{9} 	imes \frac{2}{5} = \frac{5}{27} + \frac{8}{45} = \frac{25 + 24}{135} = \frac{49}{135}$. The probability that they disperse without dining together is $P(D') = 1 - P(D) = 1 - \frac{49}{135} = \frac{86}{135}$.

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