# Fun with conditional probabilities

On Intrade right now, Barack Obama is trading at 75% to be the Democratic Presidential candidate.

Hillary Clinton is trading at 15% to be the Vice-Presidential candidate.

Let’s assume that one of them will be the Democratic Presidential candidate, but not necessarily the Vice-Presidential candidate.

Quick summary of handy standard notation: P(x) is shorthand for “the probability of event x”. P(x|y) is shorthand for “the conditional probability of event x given event y”, also known as “the probability of event x in the hypothetical world where y actually happens”. P(x âˆ§ y) means the probability of event x and event y.

Let A represent the event “Obama becomes the Presidential candidate”. Let B represent the event “Hillary Clinton becomes the Vice-Presidential candidate”. So a quick way to rephrase my first two sentences is: Intrade right now prices P(A) = 0.75 and P(B) = 0.15.

I want to talk about P(B|A); that is, the probability that Obama will select Hillary as VP should he actually get the nomination. We can derive the market’s expectation for this using Bayes’s Theorem, which I remember like this: P(x âˆ§ y) = P(x|y) * P(y). That is, the probability of both x and y is just the probability of x in the “hypothetical world” where y occurs — P(x|y) — times the probability of that “hypothetical world” being our world. Substitute x=A and y=B, then x=B and y=A, and remember that A âˆ§ B is the same thing as B âˆ§ A, and you get:

P(A|B) * P(B) = P(A âˆ§ B) = P(B|A) * P(A)

Re-arrange slightly:

P(B|A) = P(A|B) * P(B) / P(A)

In English, P(A|B) is the probability that Obama is the candidate in the hypothetical world where Hillary is VP candidate. But we are assuming one of them is the nominee, so this probability is 100% and P(A|B) = 1. So:

P(B|A) = P(B) / P(A) = 0.15 / 0.75 = 0.20.

Thus, the market thinks that among the possible worlds where Obama is the nominee, in 20% of those Hillary is his VP.

Now for the question. Suppose I think that is too low. Can I structure my bets on Intrade to express this view? That is, suppose I think P(B|A) is actually higher than 20%, but I have no opinion on whether P(A) or P(B) separately is too low or too high. Can I somehow make a bet directly on P(B|A) at 20%; i.e., with a payoff ratio of 4:1 if I am right?

Here is my answer. I can short $1 worth of A (Obama Presidential candidate) and go long$3 worth of B (Hillary VP candidate). So I put down $4 total. What are the possible outcomes? If A and B happen — Obama nominee and Hillary VP nominee — then my$1 short becomes worthless but my $3 long increases in value to$3 / 0.15 = $20. There is my 4:1 payoff. If A happens but B does not, then I lose the entire bet… But since I was wrong, that is appropriate. If A does not happen but B does, I make all sorts of money. Except that is impossible; if Obama is not the Presidential nominee, Hillary will be (by assumption), and therefore cannot be the VP candidate. So this case cannot happen. Finally, If A does not happen and consequently B does not happen, I win my short bet on Obama and it goes up to$4 (on Intrade, shorting Obama at 75% is actually buying not-Obama for 25%). But this is exactly enough to cover the loss on my long bet on Hillary! So if A does not happen, I get all of my money back, as if I never placed the bet at all. Which makes perfect sense… I am making a bet on B in the worlds where A actually happens. Where A does not happen, neither does my bet.

If you are still reading, congratulations. Here is a puzzle for you. Suppose I think 20% is too high in the example above, and I want to short B|A. Do I:

1. Set up a trade like the one above, but substituting P(Â¬B) = 0.85 in place of P(B)? Or…
2. Do I just figure out what the guy on the other side of my trades was doing in the example above, and do what he did (i.e., put down $3 on Obama Prez and$17 against Hillary Veep)?

These approaches give different answers, because one of them is right and the other is wrong. Which is right, and where does the reasoning for the other break down?