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If a person is healthy. he has a 20% chance of being greater than 70 years old. If a person is greater than 70 years old, he has an 85% chance of being healthy. Suppose 15% of the citizens in a city are greater than 70 years old. What is the probability that a randomly selected person is healthy?

 Aug 7, 2020
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Let's say that there are 10,000 persons in the city.

 

Since 15% of the citizens are greater than 70 years old: 

   15% of 10,000  =  0.15 x 10,000  =  1500 persons are older than 70

                                           and            8500 persons are younger than 70.

 

Since 85% of the persons older than 70 are healthy:

   85% of 1500  =  0.85 x 1500  =  1275 persons older than 70 are healthy

                                         and         225 persons older than 70 are not healthy.

 

Since 20% of the heathy person are greater than 70, this means that 80% of the healthy persons are younger

than 70.  Since 80% is 4 times as great as 20%, there are 4 times as many healthly persons younger than

70 than those older than 70:

   4 x 1275  =  5100 persons younger than 70 who are healthy.

 

Total number of healthy persons:  1275 + 5100  =  6375

Total number of persons:  10,000.

 

Probability of selecting a healthy person at random:  6375 / 10000  =  0.6375

 Aug 7, 2020

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