Conditional Probability or Independent events? There are 7 charitable donors at a gala - 6 "typical" donors and 1 "generous" donor. Every time a "typical" charitable donor is approached for money, she/he will donate (with equal probability) either \$0, \$1, \$2, \$4, or \$8. Every time the "generous" charitable donor is approached, she/he will always give \$8. You and your friend approach the same randomly selected charitable donor and both ask for money, and you both receive \$8. What is the probability that the charitable donor is the "generous" one? How should I approach this? I am fairly new at probability. The way I was thinking is that the 2 friends statement is given just for confusion because the event of selecting "Generous" donor given that you get $8 would have nothing to do with you or your friend. Am I thinking right?

Let $G$ represent the event that it was the generous donor who was approached, and $E$ represent the event of getting an \$8 contribution from the donor from two different requests. We are given that $P(G)=1/7$, $P(E|G)=1$, and we can infer that $P(E|G^C)=1/25$ (since a typical donor has five equally likely donations). From this, we can calculate $$P(E)=P(E|G)P(G)+P(E|G^C)P(G^C)=\frac {1}{7}\cdot 1+\frac{6}{7}\cdot\frac{1}{25}=\frac{31}{175}$$ and from here Bayes' Theorem says $$P(G|E)=\frac{P(E|G)P(G)}{P(E)}=\frac{(1/7)\cdot1}{31/175}=\frac{25}{31}$$