Posterior from a likelihood ratio

EasyConditional Probability & Expectation Statistics~3m

A signal fires with prior probability $0.2$ . When the signal is genuinely present, an alert triggers $90\%$ of the time; when it is absent, the alert still triggers $30\%$ of the time. You see an alert. What is the posterior probability the signal is present? Give a decimal to four places.

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Prior odds $=0.2:0.8=1{:}4$ ; build the LR $=0.9/0.3=3$ .
Multiply in odds form, then convert $3{:}4$ back to a probability $3/7$ .

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0.4286 (± 0.001)

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Asked at: Citadel, Two Sigma

Posterior from a likelihood ratio

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