Question

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.)

Answer 1

Answer:

• 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