Answer:
P(A) = 0.39
Step-by-step explanation:
We are given;
P(W|A) = 0.7
P(W|A^c ) = 0.3
We are told that 60% of the respondents said they voted for A. Thus;
P(A|W) = 60% = 0.6
Now, using the principle of drawing lots, we can be able to find the probability of the event that they are willing to participate in the exit poll which is P(W).
Thus;
P(W) = [P(W|A) × P(A)] +[P(W∣A^c) × P(A^c)]
Now, P(A^c) can be expressed as 1 - P(A)
Thus, we now have;
P(W) = [P(W|A) × P(A)] + [P(W∣A^c) × (1 - P(A)]
Plugging in the relevant values gives;
P(W) = 0.7P(A) + 0.3(1 - P(A))
P(W) = 0.7P(A) + 0.3 - 0.3P(A)
P(W) = 0.3 + 0.4P(A)
Now,using Baye's theorem, we can find an expression for P(A|W)
Thus;
P(A|W) = [P(A ∩ W)]/P(W)
This can be further expressed as;
P(A|W) = [P(A) × P(W|A)]/P(W)
Plugging in relevant values, we have;
0.6 = 0.7P(A)/(0.3 + 0.4P(A))
Cross multiply to get;
0.6(0.3 + 0.4P(A)) = 0.7P(A)
0.18 + 0.24P(A) = 0.7P(A)
0.18 = 0.7P(A) - 0.24P(A)
0.46P(A) = 0.18
P(A) = 0.18/0.46
P(A) = 0.39
