Question

Two candidates: a populist named T and a woman named C, are competing for the presidency of a small country with a population of 5 people eligible to vote (being T and C part of this population). each candidate votes for himself with certainty. One of the three people will vote for candidate C with certainty.

the other two voters, let's say people X and Y, are undecided. each has a 40% chance of voting for T (and 60% of doing so for C). In an interesting way, their votes have a dependency (they can discuss politics among themselves, after all, it is a very small country): conditioned to X voting for C, the probability that the voter Y does the same It's 2/3.

What si the probability that C wins the election ( that is, gets the most of the votes)?

Answer 1

Since, there are 5 voters, there are no possibilities of a tie.

T will win the election if both X and Y vote for T, on the other

hand C will win if either one of them vote for C.

For X and Y there are 4 possibilities:

i. X and Y both vote for C.

ii. X votes for C and Y for T.

iii. X votes for T and Y for C.

iv. X and Y both vote for T.

Now C wins if i, ii and iii happen.

We have, P(X = C) = P(Y = C) = 6/10 = 3/5

P(X = T) = P(Y = T) = 4/10 = 2/5

P(Y = C | X = C) = 2/3, P(Y = T | X = C) = 1/3

Now, for i, Probability that both X and Y vote for C

= P(X = C Y = C) = P(Y = C | X = C) P(X = C) = (2/3) * (3/5)

= 2/5 = 0.4.

Now for ii, Probability that X votes for C and Y for T

= P(X = C Y = T) = P(Y = T | X = C) P(X = C) = (1/3) " (3/5)

= 1/5 = 0.2.

Now, for iii, Probability that X votes for T and Y for C

= P(X = T Y = C) = P(X = T) P(Y = C) = (2/5) * (3/5) = 6/25

= 0.24.

Now, for iv, Probability that X and Y both vote for T

= P(X = T Y = T) = P(X = T) + P(Y = T) = (2/5) * (2/5) = 4/25

= 0.16.

Thus, Probability that C wins the election = Probability of i +

Probability of ii + Probability of iii = 0.4 + 0.2 + 0.24 = 0.84.

(Ans).