Apparently, depression significantly increases the risk of developing dementia later in life (BBC News, July 6, 2010). In a recent study it was reported that 22% of those who had depression went on to develop dementia, compared to only 17% of those who did not have depression. Suppose 10% of all people suffer from depression.

If a person has developed dementia, what is the probability that the person suffered from depression earlier in life? (Do not round intermediate calculations. Round your answer to 3 decimal places.)

If a person has developed dementia, what is the probability that the person suffered from depression earlier in life? (Do not round intermediate calculations. Round your answer to 3 decimal places.)

0.126

Explanation:

1. Determine the percent of people who developed dementia from the entire population.

a) From the percent of people who had depression:

10% * 22% = 2.2 %

b) From the number of people who did not have depression:

90% * 17% = 15.3%

c) Total percent:

2.2% + 15.3% 17.5% of all people developed dementia.

2. Probability that the person suffered from depression earlier in life, if a person has developed dementia:

percent of people who developed dementia and had depression / percent of people who developed dementia = 2.2% / 17.5% = 0.126

Note, that this can be calculated using the formula for conditional probability:

P (A/B) = P (A∩B) / P (B)

Where:

A is having suffered depression B is having developed dementia A∩B is having both suffered depression and developed dementia

P (A∩B) = 22% ₓ 10% = 2.2% = 0.022

P (B) = 22% * 10% + 17% * 90% = 17.5% = 0.175

P (A/B) = P (A∩B) / P (B) = 0.022 / 0.175 = 0.126