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Trusting a model signal

ProbabilityEasy

A stock rises tomorrow with probability $0.4$ . A model raises a bullish flag with probability $0.9$ when the stock will rise and with probability $0.2$ when it will not. The flag is raised. What is the probability that the stock rises?

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Hint

Weigh the flag probability in each case by the prior, then take the share coming from the rising case.

Solution

The joint probability of a rise and a flag is $0.4 \times 0.9 = 0.36$ .
The joint probability of no rise and a flag is $0.6 \times 0.2 = 0.12$ .
The total probability of a flag is $0.36 + 0.12 = 0.48$ .
The posterior is $0.36 \div 0.48 = \frac{3}{4}$ .
The answer is $\frac{3}{4}$ .

Answer: $\frac{3}{4}$

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