More fun with conditional probabilities

This post will be even more dull than the other one, but please skim that first anyway since I want to use the same notation.

Let A represent the event “Barack Obama becomes the Democratic nominee”.  Let B represent the event “Barack Obama becomes our next President”.  As I write, Intrade prices P(A) at 0.911 (91.1%) and P(B) at .5375.  (I am using the midpoint of the bid and ask, but the result does not change much if you use the “last” price.) Assuming the Intrade punters are pricing these contracts correctly, what then is P(B|A); that is, the probability that Obama will win the Presidency should he become the nominee?

Bayes’s theorem says P(B|A) = P(A|B) * P(B) / P(A).  But P(A|B) is the probability that Obama becomes the nominee in the hypothetical world where he also becomes President, which is 100%.  Therefore:

P(B|A) = P(B) / P(A) = .5375 / .911 =  .5900

So Intrade says Obama would have a 59% chance of winning the general election.

Now, let C represent the event “Hillary Clinton becomes Democratic nominee”.  Let D represent the event “Hillary Clinton becomes our next President.”  As I write, Intrade is pricing P(C) = .0865 and P(D) = .0795.  Therefore:

P(D|C) = P(D) / P(C) = .0795 / 0.865 = 0.9191

So Intrade thinks Hillary would have a 92% chance of beating McCain.

Now, you can argue about these numbers — and place the corresponding bets on Intrade if you’re so smart — but the general picture is probably accurate, and I suspect it is not lost on the Democratic leadership.  If I were a Democratic superdelegate…  Hey, stop laughing and bear with me!  If I were a Democratic superdelegate, many things would enter into my decision, but winning in November would be high on the list.  I would look at these numbers (or the same conclusion drawn from a dispassionate tallying on an electoral map) and be very concerned.

Bottom line: I believe Russert, Stephanopoulos, and so on are wrong.  This race is not over.  I am certainly no fan of Hillary Clinton, but the media bias against her has been noticeable for a while.

Addendum (10:00 next day)

Strictly speaking, it is a fallacy to derive specific probabilities from a prediction market, since Intrade does not assign a single price to any particular contract; it assigns two prices.  A precise analysis must use the bid price and the ask price to derive a range of implied probabilities for each event. For example, this morning Intrade provides the following prices:

  • (A) Obama nominee: bid 90.5, ask 90.7
  • (B) Obama president: bid 55.4, ask 56.0
  • (C) Clinton nominee: bid 8.5, ask 8.9
  • (D) Clinton president: bid 5.7, ask 6.1

This means the range of implied probabilities for “Obama nominee”, for instance, is between 90.5% and 90.7%.  Note that some of these numbers have moved significantly since yesterday.

To derive a range for P(B|A), we just divide the bid for B by the ask for A, and then vice versa.  Similarly for P(D|C).  Thus, the market now says Obama’s chances against McCain would be between 61.1% and 61.9%, while Hillary’s would be between 64.0% and 71.8%.  Dang, I knew I should have shorted her last night!  (You can quote me on that.)

This still suggests she would be the stronger candidate in the general election, but not by as wide a margin as implied by yesterday’s numbers.  So maybe Russert et. al. are right after all. :-)

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