Use Bayes' rule to find the indicated probability.
-In Cumberland County,55% of registered voters are Democrats,30% are Republicans and 15% are independent. During a recent election,35% of the Democrats voted,65% of the Republicans voted,and 75% of the
Independent voted.What is the probability that someone who voted is a Democrat?
A)0.615
B)0.55
C)0.45
D)0.385

Question

Quizplus · Accepted Answer

We need to find the probability of someone being a Democrat given that they voted, which can be represented as P(D|V). We can use Bayes' rule to calculate this:

P(D|V) = P(V|D) * P(D) / P(V)

We know that P(D) = 0.55, P(R) = 0.3, and P(I) = 0.15. We also know that:
- P(V|D) = 0.35, meaning 35% of Democrats voted
- P(V|R) = 0.65, meaning 65% of Republicans voted
- P(V|I) = 0.75, meaning 75% of Independents voted

To calculate P(V), we can use the Law of Total Probability:
P(V) = P(V|D) * P(D) + P(V|R) * P(R) + P(V|I) * P(I)

P(V) = 0.35 * 0.55 + 0.65 * 0.3 + 0.75 * 0.15
P(V) = 0.445

Now we can use Bayes' rule to find P(D|V):
P(D|V) = 0.35 * 0.55 / 0.445
P(D|V) = 0.385

Therefore, the probability that someone who voted is a Democrat is 0.385 or 38.5%. The best choice is D, with the correct probability.

Use Bayes' Rule to Find the Indicated Probability