Question

An investor wants to make a decision which depends on the state of a country's economy. The probability that the economy will be strong, P(A) is 0.80, while the probability that it will be weak, P(A'), is 0.20 (i.e. these are prior probabilities). She wants to revise these probabilities by consulting an economic forecaster who does not have a perfect forecasting record. Let F1 be the event that he will forecast a strong economy. Based on his past record, we are told that P(F1|A) = 0.40 and P(F1|A') = 0.73. Using this information, compute the revised probabilities. That is, compute P(A|F1) Blank 1 and P(A'|F1) Blank 2. Give your answers to four decimal places.

Answer 1

Answer on Question #40661, Math, Statistics and probability

An investor wants to make a decision which depends on the state of a country's economy. The probability that the economy will be strong, $P(A)$ is 0.80, while the probability that it will be weak, $P(A')$, is 0.20 (i.e. these are prior probabilities). She wants to revise these probabilities by consulting an economic forecaster who does not have a perfect forecasting record. Let F1 be the event that he will forecast a strong economy. Based on his past record, we are told that $P(F1|A) = 0.40$ and $P(F1|A') = 0.73$. Using this information, compute the revised probabilities. That is, compute $P(A|F1)$ Blank 1 and $P(A'|F1)$ Blank 2. Give your answers to four decimal places.

Solution

The probability that the economy will be strong,

$$P(A) = 0.80.$$

The probability that the economy will be weak

$$P(A') = 0.20.$$

Let $F_{1}$ be the event that he will forecast a strong economy. Based on his past record, we are told that

$$P(F_{1}|A) = 0.40 \text{ and } P(F_{1}|A') = 0.73.$$

Apply Bayes' Theorem:

$$P(A|F_{1}) = \frac{P(F_{1}|A)P(A)}{P(F_{1}|A)P(A) + P(F_{1}|A')P(A')} = \frac{0.40 \cdot 0.80}{0.40 \cdot 0.80 + 0.73 \cdot 0.20} = 0.6867,$$ $$P(A'|F_{1}) = \frac{P(F_{1}|A')P(A')}{P(F_{1}|A)P(A) + P(F_{1}|A')P(A')} = \frac{0.73 \cdot 0.20}{0.40 \cdot 0.80 + 0.73 \cdot 0.20} = 0.3133.$$