# Conditional Probabilities are Hard to Determine

One tempting thing to do with market-based probabilities of victory is to determine the conditional probability of candidate winning the general election; simply divide the probability of victory of getting the nomination by the probability of winning the general election! But, there are two serious problems with that: (1) most candidates have too much uncertainty in the probability relative to the absolute value, making the statistics poorly defined (addressed in this paper I wrote with David Pennock) (2) candidates do not win nominations independently of their probability of victory in the general election.

Only five candidates approach a 5% likelihood of winning the general election, so I will hold this article to them: Hillary Clinton (D), Marco Rubio, Donald Trump, Ted Cruz, and Jeb Bush (R).

The quick first pass is to go to PredictWise’s Presidential Election – Winner page and divide through the nomination column by the winner column. Here is what you get: Hillary Clinton 61%, Marco Rubio 42%, Donald Trump 38%, Ted Cruz 36%, and Jeb Bush 39%.

A second pass creates some bounds to this point-estimate by using the marginal bid and ask in each market within Betfair. By dividing the ask of the nomination over the bid of the winner market, we get the highest possible conditional probability. And, dividing the bid of the nomination over the ask of the winner market, we get the lowest possible conditional probability. This gives us: Hillary Clinton 59-61%, Marco Rubio 45-50%, Donald Trump 35-49%, Ted Cruz 40-49%, and Jeb Bush 43-51%. Three of the five point-estimates are outside of the ranges! This is because of the aggregating, rounding, and normalizing that takes the raw pricing data make the PredictWise values. This process inflates the probability of victory for the lower tier Republican candidates relative to their probability of winning the general election.

Of course, none of this deals with the updating that would occur should a candidate win the nomination. To win the nomination promises need to be made, skills acquired, and events need to transpire. This could ultimately bend the probability of victory out of this strict conditional relationship.

One thing is clear, even with the wide ranges of probable conditional probabilities; the Democratic front-runner is significantly above 50% likely to win the election conditional on winning the nomination and the Republican candidates are not.

Finally, there is nothing in Trump's conditional probability of victory to indicate a concern that he is likely to compete for the presidency should he lose the nomination.