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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 candidates 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)?

I tried different solutions but all of them seem to be wrong. for C to win she has three ways, either voter3 votes for her, or voter4 votes for her or voter 3 and 4 vote for her.

$P(3C) = P(3C|4C).P(4C) + P(3C|4T).P(4T) = \frac{6}{10} *\frac{2}{3} * \frac{6}{10} + \frac{6}{10} *\frac{1}{3} * \frac{4}{10} = 0.32$

$P(4C) = P(4C|3C).P(3C) + P(4C|3T).P(3T) = \frac{6}{10} *\frac{2}{3} * \frac{6}{10} + \frac{6}{10} *\frac{1}{3} * \frac{4}{10} = 0.32$

$P(3Cand4C) =P(3C) * P(4C) = 0.32 * 0.32 = 0.1024$

$P(CWinning) = 0.32+ 0.32 + 0.1024 = 0.7424$

which is the wrong answer, but why?? does it have something to do with the two voters being dependent and the two events are not independent!

Could someone please tell me how to approach this question and what is wrong with my solution?

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    $\begingroup$ The only way $T$ wins is to get both uncertain votes, so that's the simpler thing to analyze. So, first you have to compute the probability that $\#4$ votes $T$ given that $\#3$ has voted for $T$. $\endgroup$
    – lulu
    Commented Nov 23, 2023 at 19:14
  • $\begingroup$ Note that $P(3C\text{ and }4C)$ has no reason to be $P(3C)\cdot P(4C)$ because you explicitly state that the choices are not independent. $\endgroup$ Commented Nov 23, 2023 at 19:21
  • $\begingroup$ @lulu thanks, understood now $\endgroup$
    – Moaz Nasem
    Commented Nov 23, 2023 at 19:47

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When you write $P(3C, 4C)= P(3C)P(4C)$ you implicitly use that the votes are independent. But by the problem statement, they are explicitly not!

We are given that $P(3C)=P(4C)=\frac35$ and $P(4C|3C)=\frac23\ne\frac35$. Then from $$ \frac35 = P(4C) = P(4C|3C)P(3C)+P(4C|3T)P(3T) = \frac 23\frac35+P(4C|3T)\frac 25,$$ we get $P(4C|3T)= \frac12$. Then $P(4T|3T)=\frac12$ and finally $$P(4T,3T) = P(4T|3T)P(3T)=\frac12\frac25=\frac15$$ and this is also the probability for $T$ to win the election as a whole.

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  • $\begingroup$ Thanks a lot, now I understood what I didn't understand before. $\endgroup$
    – Moaz Nasem
    Commented Nov 23, 2023 at 19:46

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