OK folks, question about problem… Colonia has only two political parties, the Wigs and the Wags. If the Wags are elected there is a 32% probability of a tax increase over the next four years. If the Wigs are elected, there is a 60% probability of a tax increase. Based on current polls there is a 20% probability that the Wags will be elected. The sum of the (unconditional) probability of a tax increase and the joint probability that the Wigs will be elected and there will be no tax increase are closest to: A) 55% B) 65% C) 75% D) 85% So I identify the following: P(tax | wag) = .32 P(tax| wig) = .60 P(wag) = .20 so P(tax) = P(tax|wag)*P(wag) + P(tax|wig)*P(wig) = .32*.2 + .6*.8 = 0.544 The question asks for us to present P(tax) + P(wig & no tax) Here is where I get confused. P(wig & no tax) = (1 - P(wag)) * (1-0.544) = 0.8*0.456 But the answer lists P(wig & no tax) as = 0.8*0.4, 0.4 being P(no tax | wig). This question seems poorly worded to be because I don’t feel as though there’s a clear indication whether we should be solving for P(wig & no tax), P(wig|no tax) (which would be ~0.70 if i’m not mistaken), or P(no tax| wig). Can someone please explain how to know what the problem is looking for? Hopefully the official exam will be more clear than this.
The answer is correct. You are asked for the joint probability (that is both have to occur) of wigs and no tax: P(A&B)=P(A)*P(B). P(wig|no tax) is a posterior probability: you observe there is no tax increase, what is the probability wigs are ellected. P(no tax|wig) is the conditional probability of no tax increase, given wigs have been ellected.
I guess the answer was D?
I had the same issue with this question, and I think I figured it out. So for anyone who’s still wondering:
When they say “…the joint probability that the Wigs will be elected and there will be no tax…,” they mean P(no tax|Wigs), or “the probability that there will be no tax, given that Wigs are elected…” They don’t mean P( Wigs + No Tax), or the probability of Wigs being elected AND no tax, as the OP says. However, in my opinion, I think the OP is right and the book is wrong. Hence, the wording in the book is misleading at best, and flat out wrong at worst.
They tell you in the question that the probability that taxes increase, given that Wigs are elected, or P(Tax|Wigs) is 60%. Therefore, the probability that taxes do not increase, given that Wigs are elected, (P(No Tax|Wigs), is 1 - 0.60 = 40%. That’s where the 0.4 comes from, at least that’s what I think. “Joint probability” should mean the separate unconditional probabilities, but it apparently doesn’t here.