Problem 56 - Algebra and Trigonometry with Analytic Geometry

Question

Life expectancy A man is 54 years old and a woman is 34 years old. The probability that the man will be alive in 10 years is \(0.74\), whereas the probability that the woman will be alive 10 years from now is \(0.94\). Assume that their life expectancies are unrelated. (a) Find the probability that they will both be alive 10 years from now. (b) Determine the probability that neither one will be alive 10 years from now. (c) Determine the probability that at least one of the two will be alive 10 years from now.

Step-by-Step Solution

Answer

(a) 0.6956; (b) 0.0156; (c) 0.9844.

Step 1: Identify Given Probabilities

The probability that the man is alive in 10 years is given as \( P(M) = 0.74 \). The probability that the woman is alive in 10 years is \( P(W) = 0.94 \). Since their life expectancies are unrelated, these events are independent.

Step 2: Calculate Probability Both are Alive

To find the probability that both the man and the woman will be alive in 10 years, multiply their independent probabilities: \ \[ P(M \cap W) = P(M) \times P(W) = 0.74 \times 0.94 \] \ Calculate to get \( P(M \cap W) = 0.6956 \).

Step 3: Calculate Probability Both are Not Alive

First, find the probabilities that each is not alive in 10 years: \( P(M') = 1 - P(M) = 0.26 \) and \( P(W') = 1 - P(W) = 0.06 \). Since they are independent: \ \[ P(M' \cap W') = P(M') \times P(W') = 0.26 \times 0.06 \] \ Calculate to find \( P(M' \cap W') = 0.0156 \).

Step 4: Calculate Probability at Least One is Alive

To find the probability that at least one of them is alive, use the complement rule: \ \[ P(M \cup W) = 1 - P(M' \cap W') = 1 - 0.0156 \] \ Thus, \( P(M \cup W) = 0.9844 \).

Key Concepts

Independent Events

In probability, independent events are events whose outcomes do not affect each other. For instance, if you flip a coin and roll a die, the outcome of the coin flip does not affect the die roll. Similarly, the life expectancies of the man and the woman in our exercise are independent, meaning one person's survival does not influence the other's.
To identify independence, check that a condition similar to this holds: If the occurrence of one event does not change the probability of another event occurring, the events are independent.
In mathematical terms, two events, A and B, are independent if the probability of both A and B occurring together is the product of their individual probabilities. This is calculated as:
\[ P(A \cap B) = P(A) \times P(B) \].
In our example, since the probability of the man surviving 10 years is 0.74, and the probability of the woman surviving 10 years is 0.94, they are considered independent events. So, when wanting to determine the probability they will both be alive, these are simply multiplied together.

Complement Rule

The complement rule is a fundamental concept in probability used to find the probability of 'not' happening. When you know the probability of an event, you can easily find the probability of its complement (the event not happening).
Mathematically, the complement rule is:

• \[ P(A') = 1 - P(A) \]

where \( A' \) represents the complement of event \( A \).
This rule is particularly helpful in problem-solving because sometimes it is easier to calculate the probability of the complement and then subtract it from 1 to get the probability of the original event.
In the exercise, to find the probability that neither the man nor the woman will be alive in 10 years, we first calculated the individual complementary probabilities: the man not being alive \( P(M') = 1 - 0.74 = 0.26 \), and the woman not being alive \( P(W') = 1 - 0.94 = 0.06 \). By multiplying these, we get the probability that both are not alive in 10 years. Moreover, using the complement rule helps solve for probabilities like at least one being alive, where we find the complement of both not surviving.

Multiplication Rule in Probability

The multiplication rule in probability is crucial when dealing with independent events. This rule allows us to determine the probability of two or more independent events happening together.
According to the multiplication rule, if events A and B are independent, then the probability of both A and B occurring is:

• \[ P(A \cap B) = P(A) \times P(B) \]

This rule applies when there is no overlap or dependency between the events.
Utilizing this rule helps to manage complex situations where we want to evaluate the occurrence of multiple independent outcomes. For instance, in the given exercise, because the probability of the man being alive and the probability of the woman being alive are independent, we apply this rule to find the probability they will both be alive in 10 years.
Starting with their individual probabilities, the exercise demonstrates the use of the multiplication rule effectively by calculating the joint likelihood of survival as \( 0.74 \times 0.94 = 0.6956 \), reflecting the idea that both are treated as separate events.