In a town it rains on 20% of the days. Rain had been forecast on 90% of the rainy days and on 10% of the dry days. A day is chosen at random. Consider the following statements:
(A) The probability that rain was forecast for the day is 0.26.
(B) If rain was forecast, the probability that it rained is \(\displaystyle\frac{9}{13}\).
(C) If rain was not forecast, the probability that it rained is \(\displaystyle\frac{1}{10}\).
(D) The probability that the forecast was wrong (rain forecast on a dry day, or no rain forecast on a rainy day) is 0.1.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A), (B) and (D) only
Let \(E_1\) and \(E_2\) be the events that the day is rainy and dry, and F the event that rain was forecast.
\(P(E_1)P(F|E_1) = 0.2\times 0.9 = 0.18\)
\(P(E_2)P(F|E_2) = 0.8\times 0.1 = 0.08\)
By the theorem of total probability:
\(P(F) = 0.18+0.08 = 0.26\)
By Bayes’ theorem:
\( \displaystyle P(E_1|F) = \frac{0.18}{0.26} = \frac{9}{13} \)
On a rainy day rain was not forecast with probability 0.1, and \(P(F') = 1-0.26 = 0.74\):
\(P(E_1)P(F'|E_1) = 0.2\times 0.1 = 0.02\)
\( \displaystyle P(E_1|F') = \frac{0.02}{0.74} = \frac{1}{37} \)
The forecast is wrong on a dry day with a rain forecast or on a rainy day without one:
\(P(\text{wrong forecast}) = 0.08+0.02 = 0.10\)
So P(forecast) = 0.26, P(rain | forecast) = \(\frac{9}{13}\), P(rain | no forecast) = \(\frac{1}{37}\) and P(wrong forecast) = 0.1.
Checking the statements
(A) The probability that rain was forecast for the day is 0.26. — Correct. 0.18 + 0.08 = 0.26.
(B) If rain was forecast, the probability that it rained is \(\frac{9}{13}\). — Correct. \(\frac{0.18}{0.26} = \frac{9}{13}\).
(C) If rain was not forecast, the probability that it rained is \(\frac{1}{10}\). — Incorrect. It is \(\frac{0.02}{0.74} = \frac{1}{37}\); 0.1 is the chance of no forecast on a rainy day.
(D) The probability that the forecast was wrong (rain forecast on a dry day, or no rain forecast on a rainy day) is 0.1. — Correct. 0.08 + 0.02 = 0.1.