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Noisy indicators

Market makingHard

An event has prior probability $0.2$ . $2$ independent indicators are observed. Each indicator is on with probability $0.7$ if the event happens and with probability $0.2$ if it does not. Both indicators turn out to be on. A contract pays $100$ if the event happens and $0$ otherwise. What is its new fair value? Give an exact fraction.

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Hint

Multiply the prior by the likelihood of the observations under each hypothesis.

Solution

Joint weight if the event happens: $0.2 \times 0.7 \times 0.7 = 0.098$ ; if not: $0.8 \times 0.2 \times 0.2 = 0.032$ .
The posterior probability is $0.098 \div 0.13 = \frac{49}{65}$ .
The fair value is $100 \times \frac{49}{65} = \frac{980}{13}$ .

Answer: $\frac{980}{13}$

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