A statistical investigation showed that adults in a particular country have an 80% chance of living to be at least 70 years old and a 50% chance of living to be at least 80 years old. What is the probability that an adult who just turned 70 will live to be 80?

Question
Probability
asked 2021-03-12
A statistical investigation showed that adults in a particular country have an 80% chance of living to be at least 70 years old and a 50% chance of living to be at least 80 years old. What is the probability that an adult who just turned 70 will live to be 80?

Answers (1)

2021-03-13
The probability of living to be at least 80 years old is dependent on the probability of living to be at least 70 years old since a person can't live to be at least 80 years old if they did not live to be at least 70 years old.
The formula for two dependent events A and B is P(A and B)=P(A)⋅P(B∣A).
Let AA be the probability that a person lives to be at least 70 years old and BB be the probability that a person lives to be at least 80 years old. You then need to find P(B∣A) since you need to find the probability that a person will live to be at least 80 years old given that they have already lived to be 70 years old. It is given that the probability of living to be at least 70 years old is 80% so P(A)=0.8. It is also given that the probability of living to be at least 80 years old is 50% so P(A and B)=0.5. Therefore:
P(A and B)=P(A)⋅P(B∣A)
0.5=0.8⋅P(B∣A)
0.5/0.8=P(B∣A)
0.625=P(B∣A)
The probability that a person will live to be at least 80 years old given that they have just turned 70 years old is then 0.625=62.5%.
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