An election probability problem I can't figure out

okimhin

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Dec 25, 2016
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The Question:
Two candidates, a demagogue named T and a woman named C, are competing for the presidency of a small country with a total voting-age population of five people. Voters 1 and 2 are both certain they'll vote for C, while voter 5 has chosen T. (Let's assume the candiates don't vote — or, if you prefer, C is voter 1 and T is voter 5.)

Voters 3 and 4 are undecided: each has a 40% probability of voting T (and a 60% probability of voting C). Interestingly, their votes are correlated (perhaps they discuss politics together — it's a small country, after all): conditional on voter 3 choosing C, there is a 2/3 chance that voter 4 will do the same.
What's the probability that C wins the election (i.e. that she wins a majority of the country's five votes)?
My thoughts:
I am thinking along the line that all five will vote for at least one of the candidates and then the probability of a T win is only if both undecided voters go to T. And the probability of a C win is 1 - Pr(T win).

For voter 3 and 4 you have CC, CT, TC and TT as possible outcomes. And I use TT as 0.4 * 0.4 = 0.16. But 1 - 0.16 is wrong. I guess that condition plays in but that only affects if voter 3 votes C and that is already a C win. I can't figure out how the condition plays in.
 
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