Question

Fifty-six percent of all American workers have a workplace retirement plan, \(68 \%\) have health insurance, and \(49 \%\) have both benefits. We select a worker at random. a) What's the probability he has neither employersponsored health insurance nor a retirement plan? b) What's the probability he has health insurance if he has a retirement plan? c) Are having health insurance and a retirement plan independent events? Explain. d) Are having these two benefits mutually exclusive? Explain.

Short answer

a) 0.25. b) 0.875. c) Not independent. d) Not mutually exclusive.

Step-by-step solution

Define Key Terms

We define two events. Let event A represent workers having a workplace retirement plan and event B represent workers having health insurance. We are given that 56% of workers have a retirement plan (P(A) = 0.56), 68% have health insurance (P(B) = 0.68), and 49% have both benefits (P(A ∩ B)=0.49).

Calculate Probability of Neither Benefit (Part a)

To find the probability that a worker has neither health insurance nor a retirement plan, we use the formula for the union of two sets. The probability of A union B is given by \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \). Substituting the given probabilities, we get \( P(A \cup B) = 0.56 + 0.68 - 0.49 = 0.75 \). The probability of neither event occurring, \( P(A^c \cap B^c) \), is therefore \( 1 - P(A \cup B) = 1 - 0.75 = 0.25 \).

Calculate Conditional Probability (Part b)

To find the probability that a worker has health insurance given that they have a retirement plan, we use the conditional probability formula: \( P(B|A) = \frac{P(A \cap B)}{P(A)} \). Substituting the known values, we have \( P(B|A) = \frac{0.49}{0.56} \approx 0.875 \).

Test for Independence (Part c)

Two events A and B are independent if \( P(A \cap B) = P(A) \times P(B) \). Here, \( P(A \cap B) = 0.49 \) while \( P(A) \times P(B) = 0.56 \times 0.68 = 0.3808 \). Since \( 0.49 eq 0.3808 \), the events are not independent.

Test for Mutual Exclusivity (Part d)

Two events are mutually exclusive if they cannot occur at the same time, i.e., if \( P(A \cap B) = 0 \). Given \( P(A \cap B) = 0.49 eq 0 \), the events are not mutually exclusive.

Key concepts

Conditional Probability

Conditional probability is used to find the probability of an event occurring, given that another event has already occurred. In our exercise, we need to determine the probability that a worker has health insurance given they already have a retirement plan. We denote this as \(P(B|A)\). The concept here is that we're looking within a specific subset of the population \, the workers who already have a retirement plan, to see how many also have health insurance.
To calculate this, we use the formula for conditional probability:

• \(P(B|A) = \frac{P(A \cap B)}{P(A)}\)

This formula divides the probability of both events happening by the probability of one event. Inserting the given values, we find \(P(B|A) = \frac{0.49}{0.56} \approx 0.875\). This means that 87.5% of workers who have a retirement plan also have health insurance. So, conditional probability sheds light on the likelihood of overlapping events within a total set.

Independence in Probability

In probability, two events are independent if the occurrence of one event does not affect the probability of the other. This means that the events don't influence each other - they're like two separate, random draws.
To check for independence between having a retirement plan (Event A) and having health insurance (Event B), we use this principle:

• Events A and B are independent if \(P(A \cap B) = P(A) \times P(B)\).

When we test this using our given data, we have \(P(A \cap B) = 0.49\) and \(P(A) \times P(B) = 0.56 \times 0.68 = 0.3808\). These values are not equal (\(0.49 eq 0.3808\)), indicating that the two events are not independent. In simple terms, having one of the benefits does influence the likelihood of having the other. This is crucial in understanding the overlap and how benefits might relate or affect each other statistically.

Mutual Exclusivity in Probability

Mutual exclusivity is a straightforward concept in probability that describes when two events cannot occur simultaneously. If events are mutually exclusive, the presence of one event means that the other cannot happen. In mathematical terms, it means:

• The intersection of events A and B \( (P(A \cap B))\) must equal zero.

For our scenario, we check if having a retirement plan and having health insurance are mutually exclusive. We know \(P(A \cap B) = 0.49\), which is not equal to zero. Therefore, these events are not mutually exclusive.
This means that it is entirely possible for workers to have both a retirement plan and health insurance. Understanding this helps in clarifying how two seemingly different benefits can be provisioned together, rather than in isolation.