# Probability

3. Presume in community 1, 52% of individuals who will vote in election favor candidate A while while 48% favor candidate B. In community 2, 46% favor candidate A and 54% favor B. Community 1 is four times as large as community 2. Prior to election, a poll of 1000 randomly selected voters is taken with 800 from community 1 and 200 coming from community 2. The poll will correctly predict that A will win if the majority of poll says they favor A

3a) What is prob. that first person polled favors A?

3b) If first person polled favors B, what is prob. that the individual comes from community 1?

3c) What is the prob. the poll predicts that B will win.

3d) Presume a second independent poll of 1000 voters (800 from community 1 and 200 from 2). What is prob. A is predicted the winner by one poll and B is predicted the winner by the other?

3e) If the 1000 voters in each poll are put together to form a poll of 2000 voters, what is prob. A is predicted to win?

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#### Solution Preview

3. Presume in community 1, 52% of individuals who will vote in election favor candidate A while while 48% favor candidate B. In community 2, 46% favor candidate A and 54% favor B. Community 1 is four times as large as community 2. Prior to election, a poll of 1000 randomly selected voters is taken with 800 from community 1 and 200 coming from community 2. The poll will correctly predict that A will win if the majority of poll says they favor A

3a) What is prob. that first person polled favors A?

probability that the person is from community 1 = 800/1000 = 0.8

probability that he is from 2 = 1 - 0.8 = 0.2

If he is from community 1, the chance that he favors A = 0.52

Thus, the chance that he is from 1 and favors A = 0.8 x 0.52 = 0.416

If he is from community 2, the chance that he favors A = 0.46

Thus, the chance that he is from 2 and favors A = 0.2 x 0.46 = 0.092

Total probability = 0.416 + 0.092 = 0.508

3b) If first person polled favors B, what is prob. that the individual comes from community 1?

Probability that he is from community 1 and favors B = 0.8 x 0.48 = 0.384

Probability that he is from community 2 and favors B = 0.2 x 0.54 = 0.108

Total probability that a person favors B = 0.384 + 0.108 = 0.492

Probability ...

#### Solution Summary

The expert determines the probability that the first person polled favors A or favors B.