Question

According to a survey on US population, the probability that a man will live seventy-five years is 0.62 and that he will live up to eighty-five years is 0.023. If Mr. Clooney is seventy-five years old, what is the probability he will survive to eighty-five years? Express the probability as a percentage. Include in your final answer the calculations necessary to arrive at the probability.

Answer 1

Answer:

3.71%

Step-by-step explanation:

To find the probability that Mr. Clooney will survive to 85 years given that he is already 75 years old, we can use conditional probability.

Let event A be that a man will live up to 75 years.

Let event B be that a man will live up to 85 years.

Given probabilities:

• P(A) = 0.62
• P(B) = 0.023

We need to find the probability that a man will live up to 85 years given that he has already lived to 75 years, so we need to find P(B | A).

The conditional probability formula is:

$\boxed{\begin{array}{c}\underline{\sf Conditional\;Probability\;}\\\\\sf P(B|A)=\dfrac{P(A \cap B)}{P(A)}\end{array}}$

Given that if a man lives to 85 years, he has also lived to 75 years, we can say:

P(A ∩ B) = P(B) = 0.023

Substitute the values into the formula:

$P(B|A) = \dfrac{0.023}{0.62} \\\\\\P(B|A) \approx 0.0371$

To express this probability as a percentage, multiply by 100:

$P(B|A) \approx 0.0371 \times 100\% = 3.71\%$

So, the probability that Mr. Clooney will survive to 85 years given that he is already 75 years old is approximately 3.71%.