Answer to Question #260635 in Statistics and Probability for Okay bbe

Question

An economics consulting firm has created a model to predict recessions. The model predicts a recession with a probability of 80% when a recession is indeed coming and with a probability of 10% when no recession is coming. The unconditional probability of falling into a recession is 20%. If the model predicts a recession, what is the probability that a recession will indeed come?

Answer 1

P(Predict recession | recession coming) = 0.80

P(Predict recession | no recession coming) = 0.10

P(recession coming) = 0.20

By using the Bayes theorem formula:

P(recession coming | Predict recession) $= \frac{P(Predict \;recession | recession \;coming) \times P(recession \;coming)}{P(Predict \;recession | recession \;coming) \times P(recession \;coming) + P(Predict \;recession | recession \; not \; coming) \times P(recession \; not \; coming)}$

P(recession not coming) = 1 -P(recession coming) = 1 -0.20 = 0.80

P(recession coming | Predict recession) $= \frac{0.80 \times 0.20}{0.80 \times 0.20 + 0.10 \times 0.80}$

$= \frac{0.16}{0.24} \\ = 0.6666$