In a large population of patients, 20% have early stage cancer, 10% have advanced stage cancer, and the other 70% do not have cancer. Six patients from this population are randomly selected.

Calculate the expected number of selected patients with advanced stage cancer, given that at least one of the selected patients has early stage cancer.

• 0.403
• 0.500
• 0.547
• 0.600
• 0.625

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

Solution: C

Step 1: Define Variables and Goal

In this problem, we are given a population of patients with different cancer statuses:

Population Characteristics

Cancer Status	Proportion	Probability ([math]p[/math])
Early Stage Cancer	20%	[math]p_E = 0.20[/math]
Advanced Stage Cancer	10%	[math]p_A = 0.10[/math]
No Cancer	70%	[math]p_N = 0.70[/math]

We randomly select 6 patients from this population. Let's define the following random variables:

• [math]X[/math]: The number of selected patients with early stage cancer.
• [math]Y[/math]: The number of selected patients with advanced stage cancer.

Our objective is to calculate the expected number of selected patients with advanced stage cancer, given that at least one of the selected patients has early stage cancer. This can be expressed as [math]\operatorname{E}(Y | X \ge 1)[/math].

Step 2: Apply the Law of Total Expectation

The law of total expectation allows us to relate the unconditional expectation of [math]Y[/math] to conditional expectations based on partitions of the sample space. We can partition the sample space into two events: [math]X=0[/math] (no early stage cancer) and [math]X \ge 1[/math] (at least one early stage cancer). The formula is:

[[math]]\operatorname{E}(Y) = \operatorname{P}(X = 0) \operatorname{E}(Y | X = 0) + \operatorname{P}(X \ge 1) \operatorname{E}(Y | X \ge 1)[[/math]]

To find our desired quantity, [math]\operatorname{E}(Y | X \ge 1)[/math], we can rearrange this equation:

[[math]]\operatorname{E}(Y | X \ge 1) = \frac{\operatorname{E}(Y) - \operatorname{P}(X = 0) \operatorname{E}(Y | X = 0)}{\operatorname{P}(X \ge 1)}[[/math]]

Now, we need to calculate each component on the right side of this equation.

Step 3: Calculate Component Probabilities and Expectations

We will calculate [math]\operatorname{E}(Y)[/math], [math]\operatorname{P}(X=0)[/math], [math]\operatorname{P}(X \ge 1)[/math], and [math]\operatorname{E}(Y | X = 0)[/math].

• Expected Number of Patients with Advanced Stage Cancer (Unconditional), [math]\operatorname{E}(Y)[/math]:

The number of patients with advanced stage cancer [math]Y[/math] follows a binomial distribution [math]B(n=6, p_A=0.10)[/math]. The expected value of [math]Y[/math] is:

[[math]]\operatorname{E}(Y) = n \cdot p_A = 6 \cdot 0.10 = 0.6[[/math]]

• Probability of No Early Stage Cancer, [math]\operatorname{P}(X=0)[/math]:

The probability that a single patient does not have early stage cancer is [math]1 - p_E = 1 - 0.20 = 0.80[/math]. Since the selections are independent, the probability that none of the 6 selected patients have early stage cancer is:

[[math]]\operatorname{P}(X=0) = (1 - p_E)^6 = (0.8)^6 = 0.262144[[/math]]

• Probability of At Least One Early Stage Cancer, [math]\operatorname{P}(X \ge 1)[/math]:

This is the complement of having no early stage cancer:

[[math]]\operatorname{P}(X \ge 1) = 1 - \operatorname{P}(X=0) = 1 - (0.8)^6 = 1 - 0.262144 = 0.737856[[/math]]

• Expected Number of Patients with Advanced Stage Cancer Given No Early Stage Cancer, [math]\operatorname{E}(Y | X = 0)[/math]:

If a selected patient does not have early stage cancer, their status can be either advanced stage cancer or no cancer. The probability of having advanced stage cancer, given that a patient does not have early stage cancer, is given by the conditional probability [math]P(\text{Advanced} | \text{not Early})[/math]. Assuming "early stage cancer" and "advanced stage cancer" are mutually exclusive categories for a single patient:

[[math]]P(\text{Advanced} | \text{not Early}) = \frac{P(\text{Advanced and not Early})}{P(\text{not Early})} = \frac{P(\text{Advanced})}{1 - P(\text{Early})}[[/math]]

Using the given probabilities:

[[math]]P(\text{Advanced} | \text{not Early}) = \frac{0.10}{1 - 0.20} = \frac{0.10}{0.80} = \frac{1}{8} = 0.125[[/math]]

Given that none of the 6 selected patients have early stage cancer ([math]X=0[/math]), each of these 6 patients independently has a [math]0.125[/math] probability of having advanced stage cancer. Therefore, the conditional expected number of patients with advanced stage cancer is:

[[math]]\operatorname{E}(Y | X = 0) = n \cdot P(\text{Advanced} | \text{not Early}) = 6 \cdot \frac{1}{8} = \frac{6}{8} = 0.75[[/math]]

Step 4: Calculate the Conditional Expectation

Now, we substitute all the calculated values into the rearranged formula from Step 2:

[[math]]\operatorname{E}(Y | X \ge 1) = \frac{\operatorname{E}(Y) - \operatorname{P}(X = 0) \operatorname{E}(Y | X = 0)}{\operatorname{P}(X \ge 1)}[[/math]]

Substituting the numerical values:

[[math]]\operatorname{E}(Y | X \ge 1) = \frac{0.6 - (0.262144) (0.75)}{0.737856}[[/math]]

[[math]]\operatorname{E}(Y | X \ge 1) = \frac{0.6 - 0.196608}{0.737856}[[/math]]

[[math]]\operatorname{E}(Y | X \ge 1) = \frac{0.403392}{0.737856}[[/math]]

[[math]]\operatorname{E}(Y | X \ge 1) \approx 0.54673[[/math]]

Rounding to three decimal places, the expected number is [math]0.547[/math].

Key Insights

• The Law of Total Expectation is a powerful tool to decompose complex conditional expectation problems into more manageable parts by partitioning the sample space.
• Understanding conditional probability is crucial. When new information is given (e.g., [math]X=0[/math]), the probabilities of other events change, which in turn affects their expected values.
• The problem implies that early stage cancer, advanced stage cancer, and no cancer are mutually exclusive states for any single patient, simplifying conditional probability calculations like [math]P(\text{Advanced} | \text{not Early})[/math].
• The expected value of a binomial distribution ([math]n \cdot p[/math]) can be applied to calculate both unconditional and conditional expected numbers of patients, provided the appropriate probability parameter is used.

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

Solution: C

Let [math]X[/math] and [math]Y[/math] represent the number of selected patients with early stage and advanced stage cancer, respectively. We need to calculate [math]\operatorname{E}(Y | X ≥ 1) [/math].

From conditioning on whether or not X ≥ 1 , we have

[[math]] \operatorname{E}(Y ) =\operatorname{P}[ X =0]\operatorname{E}(Y | X =0) + \operatorname{P}[ X ≥ 1]\operatorname{E}(Y | X ≥ 1) . [[/math]]

Observe that [math]\operatorname{P}[X=0] = (1-0.2)^6 = (0.8)^6 [/math], [math]\operatorname{P}[X \geq 1] = 1- \operatorname{P}[X = 0] = 1-(0.8)^6[/math], and [math]\operatorname{E}[Y] = 6(0.1) = 0.6 [/math]. Also, note that if none of the 6 selected patients have early stage cancer, then each of the 6 selected patients would independently have conditional probability [math]\frac{0.1}{1-0.2} = \frac{1}{8}[/math] of having late stage cancer, so [math]\operatorname{E}[Y | X = 0] = 6(1/8) = 0.75 [/math].

Therefore

[[math]] \operatorname{E}(Y | X \geq 1) = \frac{\operatorname{E}(Y) − \operatorname{P}[ X= 0]\operatorname{E}(Y | X= 0)}{\operatorname{P}[X \leq 1]} = \frac{0.6 − (0.8)6 (0.75)}{1-(0.8)^6} = 0.547. [[/math]]

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