Data 605 - Probability section 4.1, question 27

Heather Geiger

Question

In London, half of the days have some rain. The weather forecaster is correct 2/3 of the time, i.e., the probability that it rains, given that she has predicted rain, and the probability that it does not rain, given that she has predicted that it won’t rain, are both equal to 2/3. When rain is forecast, Mr. Pickwick takes his umbrella. When rain is not forecast, he takes it with probability 1/3. Find:

1. The probability that Pickwick has no umbrella, given that it rains.

2. The probability that he brings his umbrella, given that it doesn’t rain.

Answer

We are given that the PPV and NPV are both equal to 2/3.

Solving for one of the two variables in the PPV and NPV formulas, we obtain the following expressions:

FP = 0.5TP, or TP = 2FP

FN = 0.5TN, or TN = 2FN

Plug these into:

TP + FN = 0.5

FP + TN = 0.5

To get:

TP + 0.5TN = 0.5

0.5TP + TN = 0.5

To get that TP = (1/3), TN = (1/3), FP = (1/6), and FN = (1/6).

This gives the following overall probabilities when combined with Pickwick’s umbrella policy.

overall_probabilities = data.frame(Forecast = rep(c("Yes","No"),each=4),
    Actual = rep(c("Yes","No"),each=2,times=2),
    Umbrella = rep(c("Yes","No"),times=4),
    Probability.out.of.18 = c(6,0,3,0,1,2,2,4),
    stringsAsFactors=FALSE)
overall_probabilities

##   Forecast Actual Umbrella Probability.out.of.18
## 1      Yes    Yes      Yes                     6
## 2      Yes    Yes       No                     0
## 3      Yes     No      Yes                     3
## 4      Yes     No       No                     0
## 5       No    Yes      Yes                     1
## 6       No    Yes       No                     2
## 7       No     No      Yes                     2
## 8       No     No       No                     4

The probability that Pickwick has no umbrella, given that it rains, is given by:

numerator = sum(overall_probabilities$Probability.out.of.18[overall_probabilities$Umbrella == "No" & overall_probabilities$Actual == "Yes"])
denominator = sum(overall_probabilities$Probability.out.of.18[overall_probabilities$Actual == "Yes"])
numerator

## [1] 2

denominator

## [1] 9

2/9

The probability that he brings his umbrella, given that it doesn’t rain, is given by:

numerator = sum(overall_probabilities$Probability.out.of.18[overall_probabilities$Umbrella == "Yes" & overall_probabilities$Actual == "No"])
denominator = sum(overall_probabilities$Probability.out.of.18[overall_probabilities$Actual == "No"])
numerator

## [1] 5

denominator

## [1] 9

5/9