Problem 55 - Basic Statistics for Business and Economics with Student

Question

Brooks Insurance, Inc., wishes to offer life insurance to men age 60 via the Internet. Mortality tables indicate the likelihood of a 60-year-old man surviving another year is .98. If the policy is offered to five men age 60 : a. What is the probability all five men survive the year? b. What is the probability at least one does not survive?

Step-by-Step Solution

Answer

a. 0.9039 b. 0.0961

Step 1: Calculate the Probability of All Five Surviving

To find the probability that all five 60-year-old men survive for a year, use the probability of one man surviving, which is 0.98. Since the survival events of the five men are independent, the probability that all five survive is the product of the probabilities for each man:\[P( ext{All survive}) = 0.98^5.\]

Step 2: Compute the Probability for All Five Survival

Calculate the probability using the formula from Step 1:\[P( ext{All survive}) = 0.98^5 \approx 0.9039.\]

Step 3: Find the Probability of At Least One Not Surviving

The probability that at least one man does not survive is the complement of all five surviving. Use the complement rule:\[P( ext{At least one does not survive}) = 1 - P( ext{All survive}).\]

Step 4: Compute the Complement Probability

Now substitute the value obtained in Step 2 into the complement formula:\[P( ext{At least one does not survive}) = 1 - 0.9039 = 0.0961.\]

Key Concepts

Independent Events

In probability theory, understanding the concept of independent events is crucial. Independent events are those whose outcomes do not affect each other.
For instance, if you flip a coin twice, the result of the first flip does not influence the second flip. This is the same in our insurance exercise where each 60-year-old man's survival for a year is independent of others.
Consider each man's survival as an independent event with its probability of 0.98. Therefore, the total probability of all men surviving involves multiplying the probabilities of each individual, independent event.
Thus, independent events simplify the calculation as impacts are not carried over from one event to another.

Complement Rule

The complement rule is one of the basic principles in probability theory. It states that the probability of an event not occurring is 1 minus the probability of the event occurring.
This rule comes in handy when determining probabilities that are not directly calculated.
In our example, to find the probability that at least one man does not survive the year, we use the complement of all five men surviving.
This is represented mathematically as:

• The probability that all survive: \( P(\text{All survive}) = 0.98^5 \)
• The complement probability: \( P(\text{At least one does not survive}) = 1 - P(\text{All survive}) \)

Using this rule makes difficult, indirect calculations easier to solve.

Mortality Tables

Mortality tables, also known as life tables, are charts that show the probability of death for a given age.
They are an essential tool in life insurance and are used to estimate life expectancy and survival probability.
These tables help actuaries and insurance companies predict the chances of survival for individuals, which is indispensable for calculating insurance premiums and risk assessment.
In our exercise, a mortality table gives the survival probability for 60-year-old men as 0.98.
This data serves as the basis for our probability calculations in the insurance scenario as it provides insight into expected outcomes based on historical data.

Survival Probability

Survival probability is a critical component when considering life insurance and evaluating risks.
It refers to the likelihood that an individual will survive over a specified period.
In our case, the survival probability of a 60-year-old man for one year is 0.98, which indicates high odds of living through the year.
This figure is a vital input for calculating relevant insurance probabilities.
When considering multiple people, we multiply individual probabilities because the events are independent.
Calculating survival probabilities allows insurance companies to create policies that balance risk and ensure financial solvency.