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Suppose a voter poll is taken in three states. In state A, 50% of voters support the liberal candidate, in state B, 60% of the voters support the liberal candidate, and in state C, 35% of the voters support the liberal candidate. Of the total population of the three states, 40% live in state A, 25% live in state B, and 35% live in state C. Given that a voter supports the liberal candidate, what is the probability that she lives in state B?

Solution:

A(0.4)B(0.25)C(0.35)
Liberal0.50.60.35

(1)   \begin{align*}P(State B | Liberal) & = \frac{P(leberal \cap State B)}{P(leberal)} \\& = \frac{0.25 \times 0.6}{ (0.4 \times 0.5 + 0.25 \times 0.6 + 0.35 \times 0.35)} \\& = \frac{0.15}{ 0.2 + 0.15 + 0.1225} \\& = 0.32\end{align*}

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