Question

Recently, Hector ran for president of the local school board. Of those who voted in the election, 80% had a high school diploma, and the other 20% did not. Hector got 75% of the vote of those with high school diplomas, while he got only 25% of the vote of those without high school diplomas.

Let D denote the event that a randomly chosen voter (in the school board election) had a high school diploma and D(overbar) denote the event that a randomly chosen voter did not have a high school diploma. Let H denote the event that a randomly chosen voter voted for Hector and H(overbar) denote the event that a randomly chosen voter did not vote for Hector. Fill in the probabilities to complete the tree diagram below, and then answer the question that follows. Do not round any of your responses.

P(H|D) = ?

P(D∩H) = ?
P(D) = ?

P(DHB(overbar)) = ?

P(H(overbar))|D) = ?

P(H|(D(oerbar)) = 0.25

P(D(overbar)∩H) = ?

P(D(overbar)) = 0.2

P(D(overbar)∩H(overbar)) = ?

P(H(overbar)|(D(overbar)) = 0.75

What is the probability that a randomly chosen voter did not vote for hector?

Answer 1

P(H|D) = 0.75

P(D) = 0.80

P(D H) = P(HID) P(D) = 0.75 . 0.80 = 0.60

Pl HID)-1-P(HID) = 1-0.75-0.25

P(HID) = 0.25

P(D) 1-P(D) 1-0.80 0.20

P(H D) = P(H/D)P(D) = 0.25 , 0.20 = 0.05

Pl HID)-1-P(HID) = 1-0.25-0.75

$P(\bar{H}\cap \bar{D})=P(\bar{H}|\bar{D})P(\bar{D})=0.75\cdot 0.20=0.15$

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0.25 -0,80

So the probability that a randomly chosen voter did not vote for hector is

P(H) = P(H D) + P(H D) = 0.15 + 0.20 = 0.35