In a group of health insurance policyholders, 20% have high blood pressure and 30% have high cholesterol. Of the policyholders with high blood pressure, 25% have high cholesterol. A policyholder is randomly selected from the group.

Calculate the probability that a policyholder has high blood pressure, given that the policyholder has high cholesterol.

• 1/6
• 1/5
• 1/4
• 2/3
• 5/6

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

Step 1: Define Events and State Given Probabilities

Let [math]B[/math] be the event that a policyholder has high blood pressure, and [math]C[/math] be the event that a policyholder has high cholesterol. We are given the following probabilities:

Given Probabilities

Event	Probability
High Blood Pressure	[math]P(B) = 0.20[/math]
High Cholesterol	[math]P(C) = 0.30[/math]
High Cholesterol given High Blood Pressure	[math]P(C | B) = 0.25[/math]

Step 2: Recall the Formula for Conditional Probability

We want to calculate the probability that a policyholder has high blood pressure, given that they have high cholesterol. This is denoted as [math]P(B | C)[/math]. The general formula for conditional probability is:

[[math]]P(B | C) = \frac{P(B \cap C)}{P(C)}[[/math]]

To use this formula, we first need to find the joint probability [math]P(B \cap C)[/math], which represents the probability that a policyholder has both high blood pressure and high cholesterol.

Step 3: Calculate the Joint Probability of Both Events

The joint probability [math]P(B \cap C)[/math] can be found using the multiplication rule for probabilities. This rule states that [math]P(B \cap C) = P(C | B) P(B)[/math]. We are given [math]P(C | B) = 0.25[/math] and [math]P(B) = 0.20[/math]. Substituting these values into the formula:

[[math]]P(B \cap C) = 0.25 \times 0.20 = 0.05[[/math]]

Thus, the probability that a policyholder has both high blood pressure and high cholesterol is 0.05.

Step 4: Substitute Values and Calculate the Final Probability

Now we have all the necessary components to calculate [math]P(B | C)[/math]. We use the formula from Step 2 and the calculated joint probability from Step 3:

[[math]]P(B | C) = \frac{P(B \cap C)}{P(C)}[[/math]]

Substitute the calculated [math]P(B \cap C) = 0.05[/math] and the given [math]P(C) = 0.30[/math]:

[[math]]P(B | C) = \frac{0.05}{0.30}[[/math]]

To simplify the fraction, we can multiply the numerator and denominator by 100:

[[math]]P(B | C) = \frac{5}{30} = \frac{1}{6}[[/math]]

Therefore, the probability that a policyholder has high blood pressure, given that they have high cholesterol, is [math]1/6[/math].

Key Insights

• Understanding and correctly applying the definition of conditional probability, [math]P(A|B) = P(A \cap B) / P(B)[/math], is fundamental.
• The multiplication rule for probabilities, [math]P(A \cap B) = P(A|B)P(B)[/math], is essential for calculating joint probabilities when conditional probabilities are provided.
• This problem implicitly uses a form of Bayes' Theorem, which combines the definition of conditional probability and the multiplication rule to find [math]P(B|C)[/math] from [math]P(C|B)[/math]: [math]P(B|C) = \frac{P(C|B)P(B)}{P(C)}[/math].
• Clearly defining events and tabulating given probabilities helps organize information and simplifies the problem-solving process.

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

Solution: A

Let B be the event that the policyholder has high blood pressure and C be the event that the policyholder has high cholesterol. We are given [math]\operatorname{P}(B) = 0.2, \operatorname{P}(C) = 0.3,[/math] and [math]\operatorname{P}(C | B) = 0.25 [/math]. Then,

[[math]] \operatorname{P}(B | C) = \frac{\operatorname{P}(B \cap C)}{\operatorname{P}(C)} = \frac{\operatorname{P}(C |B) \operatorname{P}(B) }{\operatorname{P}(C)} = \frac{0.25(0.2)}{0.3} = 1/6. [[/math]]

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