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
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• 2/3

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

Step 1: Define Events and Given Probabilities

Let's define the events for clarity:

• S: The event that a male is a smoker.
• C: The event that a male has a blood circulation problem.

We are given the following probabilities and relationships:

• The probability that a randomly chosen male has a blood circulation problem is 0.25: [math]P[C] = 0.25[/math].
• Males who have a blood circulation problem are twice as likely to be smokers as those who do not: [math]P[S | C] = 2 P[S | C^c][/math].

We can also deduce the probability of not having a circulation problem:

[[math]]P[C^c] = 1 - P[C] = 1 - 0.25 = 0.75[[/math]]

Step 2: Apply Bayes' Theorem and Simplify

We want to calculate the probability that a male has a blood circulation problem, given that he is a smoker, which is [math]P[C | S][/math]. We apply Bayes' Theorem, which states:

[[math]]P[C | S] = \frac{P[S | C] P[C]}{P[S]}[[/math]]

To find [math]P[S][/math], we use the Law of Total Probability:

[[math]]P[S] = P[S | C] P[C] + P[S | C^c] P[C^c][[/math]]

Substituting the expression for [math]P[S][/math] into Bayes' Theorem, we get the general form:

[[math]]P[C | S] = \frac{P[S | C] P[C]}{P[S | C] P[C] + P[S | C^c] P[C^c]}[[/math]]

Now, we substitute the given relationship [math]P[S | C] = 2 P[S | C^c][/math] into the equation and simplify by factoring out [math]P[S | C^c][/math] from the numerator and denominator:

[[math]]\begin{align*} P[C | S] &= \frac{(2 P[S | C^c]) P[C]}{(2 P[S | C^c]) P[C] + P[S | C^c] P[C^c]} \\ &= \frac{P[S | C^c] (2 P[C])}{P[S | C^c] (2 P[C] + P[C^c])} \\ &= \frac{2 P[C]}{2 P[C] + P[C^c]} \end{align*}[[/math]]

Step 3: Substitute Numerical Values and Calculate

We substitute the known values [math]P[C] = 0.25[/math] and [math]P[C^c] = 0.75[/math] into the simplified formula derived in Step 2:

[[math]]\begin{align*} P[C | S] &= \frac{2(0.25)}{2(0.25) + 0.75} \\ &= \frac{0.50}{0.50 + 0.75} \\ &= \frac{0.50}{1.25} \\ &= \frac{50}{125} \\ &= \frac{2}{5} \end{align*}[[/math]]

Thus, the probability that a male has a blood circulation problem, given that he is a smoker, is [math]\frac{2}{5}[/math].

Key Insights

• Bayes' Theorem is fundamental for calculating conditional probabilities in situations where the "reverse" conditional probability is known or can be derived.
• The Law of Total Probability is essential for computing marginal probabilities of an event by considering all mutually exclusive and exhaustive conditions.
• Recognizing and substituting relationships between conditional probabilities (e.g., [math]P[A|B] = k \cdot P[A|B^c][/math]) can significantly simplify Bayes' Theorem expressions, often leading to cancellation of unknown terms.
• Clearly defining events and listing all given probabilities and relationships is the crucial first step in solving probability problems.
• The concept of complementary events, [math]P[A^c] = 1 - P[A][/math], is frequently used in probability calculations.

This article was generated by AI and may contain errors. If permitted, please edit the article to improve it.

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.