The probability that a randomly chosen male has a blood circulation problem is 0.25. Males who have a blood circulation problem are twice as likely to be smokers as those who do not have a blood circulation problem.

Calculate the probability that a male has a blood circulation problem, given that he is a smoker.

• 1/4
• 1/3
• 2/5
• 1/2
• 2/3

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: C

Let:

[math]S[/math] = Event of a smoker

[math]C[/math] = Event of a circulation problem

Then we are given that [math]\operatorname{P}[C] = 0.25[/math] and [math]\operatorname{P}[S | C] = 2 \operatorname{P}[S | C^c] [/math].

Now applying Bayes’ Theorem, we find that

[[math]] \begin{align*} \operatorname{P}[C | S] &= \frac{\operatorname{P}[ S | C ]\operatorname{P}[C ]}{\operatorname{P}[ S | C ]\operatorname{P}[C ] + \operatorname{P}[ S | C ]( \operatorname{P}[C^c ])}\\ &= \frac{2 \operatorname{P}[ S | C^c ]\operatorname{P}[C ]}{2 \operatorname{P}[ S |C^c ]\operatorname{P}[C ] + \operatorname{P}[ S |C^c ](1 − \operatorname{P}[C ])} \\ &= \frac{2(0.25)}{2(0.25) + 0.75} \\ &= \frac{2}{2+3} \\ &= \frac{2}{5}. \end{align*} [[/math]]

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.