If P(high) = 0.31, P(low) = 0.69, P(favorable | high) = 0.86, and P(unfavorable | low) = 0.65, then what is P(favorable)?
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To find the probability of a favorable outcome, we need to use the law of total probability. This law states that the probability of an event is the sum of the probabilities of that event occurring in each of the mutually exclusive scenarios.
In this case, a favorable outcome can occur in two mutually exclusive scenarios: when the probability is high and when the probability is low.
So, we can calculate P(favorable) as follows:
P(favorable) = P(favorable | high) * P(high) + P(favorable | low) * P(low)
We know P(favorable | high) and P(high), but we don't know P(favorable | low). However, we do know P(unfavorable | low), and since the probabilities of favorable and unfavorable are complementary (they add up to 1), we can find P(favorable | low) as follows:
P(favorable | low) = 1 - P(unfavorable | low) = 1 - 0.65 = 0.35
Now we can calculate P(favorable):
P(favorable) = P(favorable | high) * P(high) + P(favorable | low) * P(low) = 0.86 * 0.31 + 0.35 * 0.69 = 0.2666 + 0.2415 = 0.5081
So, P(favorable) = 0.5081 or approximately 50.81%.
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