### Some quick electoral prediction math

10/20/2015 12:17:00 PM
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Jim Tankersley has some numbers from online prediction markets on the 2016 presidential election, suggesting that Hillary Clinton has roughly 50-50 odds of winning the presidency.

Do the prediction market numbers make sense? A little probability analysis:
Let $A$ denote the event where Clinton wins the nomination, and $B$ denote that a democratic nominee wins the presidency. Tankersley then provides the following probabilities from the prediction markets:
\begin{align} p \left( A \right) &=0.77 \\ p \left( B \right) &=0.55 \\ p \left( A \cap B \right) &=0.47 \end{align} Thus prediction markets think that if Clinton is nominated, theres a $$p \left( B \vert A \right)=\frac{p \left( A \cap B \right)}{p \left( A \right)}=\frac{0.47}{0.77}=0.61$$ chance of her beating the GOP candidate.

Note that $$p \left( A \right) p \left( B \right)=0.42 \lt p \left( A \cap B \right) =0.47$$ so market participants do think that who gets nominated matters for which party wins the White House, and they think Clinton has a better shot than all the other democrats combined. By what margin though?

It helps me to write it out. So let $\bar{A}$ be the complement of $A$, that is, the event that someone other than Clinton wins the nomination. The two are mutually exclusive complements so $$p \left( \left( A \cap B \right) \cup \left(\bar{A} \cap B \right)\right)=p\left( A \cap B \right) + p \left(\bar{A} \cap B \right)=p \left( B \right)=0.55$$ which tells us that the entire row of Ptolemies1 together have a probability of just $p \left(\bar{A} \cap B \right)=0.08$ of winning the presidency, despite the 0.23 probability that one of them will be nominated. So based on prediction market figures, if the democrats nominate a Ptolemy, he'll have a $$p \left( B \vert \bar{A} \right)=\frac{p \left( \bar{A} \cap B \right)}{p \left( \bar{A} \right)}=\frac{0.08}{0.23}=0.35$$ chance of winning the presidency.

So Clinton, according to the people who bet on this stuff, is not-quite twice as likely to win the general election if nominated.
1. Based on the democratic debate, I've started referring to all of the non-Clinton candidates collectively as the Ptolemies. This explains the reference.