Question

1. In the Legal Eagle law firm, 73% of the lawyers are male while 27% are female. Of the male lawyers, 42% are corporate lawyers. Of the female lawyers, 28% are corporate lawyers.

If a corporate lawyer is chosen at random, calculate the probability that they are female. Give your answer as a decimal to 2 decimal places.

Probability =

2. A test for a particular illness that affects 3% of the general population has the following properties:

The test will show a positive in 97% of cases where the patient has the illness and a negative in 3% of such cases.

The test will show a false positive in 6% of patients who do not have the illness and a negative result for 94% of patients who do not have the illness.

Note that a positive result means that the test indicates that the patient has the illness. A negative result means that the test indicates that the patient doesn't have the illness.

A patient has just been tested and shows a positive result. Assuming all other things being equal, and based on the above information, find the probability that the patient does not have the illness. Give your answer as a decimal to 2 decimal places.

Probability that patient does not have illness =

3. An administrator who works at a train station has to monitor the drinks vending machine. When the machine gets low on stock, a warning light appears and the administrator is meant to contact the drinks supplier to refill the machine. However, the light is not always accurate. In particular, it will occassionally come on even when the stock is not low. The administrator knows the following probabilities:

On average the vending machine is low on stock on 2 out of every 100 days. That is, the probability of the vending machine being low on a random day is 0.02.

The warning light is very accurate when it comes to positively reporting low stock. Specifically, if the stock is low, the probability that the warning light will come on is 1.

However, the warning light also occasionally comes on even if the stock is not low - although this is rare. The probability that the light will come on during the day when the stock isn't low is 0.002.

The administrator notices that the light has just appeared on the machine.

Based on this information, calculate the probability that the stock is low in the machine given that the warning light has just come on. Give your answer as a decimal to 3 decimal places.

P(stock is low|warning light is on) =

Answer 1

Question 1:

Here, we are given that:

P( male ) = 0.73 and P( female ) = 0.27
P( corporate lawyers | male ) = 0.42
P( corporate lawyers | female ) = 0.28

Using law of total probability, we get:

P( corporate lawyer ) = P( corporate lawyers | male )P(male ) + P( corporate lawyers | female ) P( female )

P( corporate lawyer ) = 0.42*0.73 + 0.28*0.27 = 0.3822

Now using Bayes theorem, we have:

P( female | corporate lawyer ) = P( corporate lawyers | female ) P( female ) / P( corporate lawyer)

P( female | corporate lawyer ) =  0.28*0.27 / 0.3822

P( female | corporate lawyer ) = 0.1978

Therefore 0.20 is the required probability here. ( rounded to 2 decimal places )