# a question about relative probability

The question is very simple and I think the professor intended to test if we understand $P(A\cup B)\geq P(A)$. Below is the full question:

Lance is 29. He attended Michigan State University and majored in business. He obtained a 2.8 GPA. Now he runs a small company doing web-based marketing and spends the rest of his time watching ESPN, playing video games, and cruising bars.

Consider the following statements about Lance:

S1. Lance partied a lot in college

S2. Lance really likes reading Jane Austen

S3. Lance likes barbeque

S4. Lance likes barbeque or is a vegetarian

S5. Lance likes barbeque and watching college football on his big-screen TV

Now consider the following four pairs of statements. For each pair, first indicate whether or not you can say anything about their relative probability based solely on the information provided in this question. If you can say something about their relative probability, next indicate what you can say about it. For example, you might be able to say that one statement has a higher probability of being true than another, or you might be able to say that the two statements in a pair have the same probability.

a. S1, S2

b. S1, S5

c. S3, S4

d. S3, S5

I highlighted c as the discussion will be focused on c.

First let $A$ be the event that Lance likes barbecue and $B$ be the event that Lance is a vegetarian.

I didn't think much when I put $P(S4)>P(S3)$. My friend reminded my that it should be $P(S4)\geq P(S3)$.

I was about to agree with her, but then I noticed that if $P(S4)= P(S3)$ is true, it means $P(A\cup B)=P(A)\Rightarrow B\subset A\Rightarrow$ If Lance is a vegetarian, then he likes barbecue$\Rightarrow$ All vegetarians like barbecue, which would be wrong from common knowledge.

Therefore, I feel $P(S4)>P(S3)$ is still the correct answer. I'm not sure if I'm logically wrong or if I'm correct. Would like some comments.

• What if $P(B)=0$? – user75138 Oct 16 '15 at 22:29
• If $P(B)$ is about the population, I can assert it is not 0. If $P(B)$ is about Lance only, then how do we interpret "The probability that Lance is a vegetarian?" It will be either 0 or 1, but never in between. – Quanfeng Zhou Oct 16 '15 at 22:42

Suppose that we add the following piece of information to the little narrative: "Lance is not a vegetarian." Then $P(S_3) = P(S_4)$, since your event $B$ has probability zero. Since it's possible from the information given that Lance is a vegetarian, it's possible that $P(S_3) = P(S_4)$.
Now, if we were asked to assign probabilities to these events based on the given narrative, it would only be reasonable to assign some probability to $B$, and moreover to assign some probability to $B \setminus A$ (Lance is a vegetarian and doesn't like barbecue). So, to anyone with common sense interpreting this narrative, $P(S_4) > P(S_3)$. (In general, it's a very bad idea to assign zero probability to anything that isn't logically impossible, because then no matter how much evidence you see to the contrary you'll never believe it.)
I think then the "right answer" is somewhat debatable, depending on how you interpret this kind of weird problem setup, but $P(S_4) \ge P(S_3)$ seems the "most" correct to me.
• I understand what you are saying. I feel that is $P(S_3\vert P(B)=0)=P(S_4\vert P(B)=0)$ rather than $P(S_3)=P(S_4)$. – Quanfeng Zhou Oct 16 '15 at 22:37
• If $P(B)$ is about Lance only, then it's either 0 or 1, which we don't know. If it's about the population, then it is not 0. – Quanfeng Zhou Oct 16 '15 at 22:39
• Not under a Bayesian interpretation of probability, which this question seems to be assuming: we can talk about the probability of Lance being a vegetarian. Most reasonable readers would give that some probability, but an unreasonable reader might assign that state zero probability, in which case they would say $P(S_3) = P(S_4)$. It's definitely true that $P(S_3 \mid \lnot B) = P(S_4 \mid \lnot B)$; some unreasonable readers might categorically assume $P(\lnot B) = 1$, e.g. because they believe that Michigan State University contains meat-eating as an absolute graduation requirement. – Dougal Oct 16 '15 at 22:42
• +1: This is clearly a subjective probability question, and so you need to cover all your bases (all possible probability assignments). Hence, you cannot rule out the (very unlikely) case that $P(B)=0$. – user75138 Oct 16 '15 at 23:24