a. When using the general addition rule, determine the probability that the age of the person obtained is either between 25 and 59 inclusive, or at least 35. How can the probability of this be computed without using the general addition rule? Choose the correct answer below.
A. Determine the number of people with ages between 25 and 59, inclusive, or at least 35, and divide that by the total number of people.
B. Determine the probability that a person chosen is between 25 and 59, inclusive, then add that to the probability that a person chosen is at least 35, then subtract from the sum the probability that the person chosen is both between 25 and 59, inclusive and at least 35.
C. Determine the probability that a person chosen is between 25 and 59, inclusive, then add that to the probability that a person chosen is at least 35.
D. Determine the number of people with ages between 25 and 59, inclusive, and at least 35, and divide that by the total number of people.
Identify the probability of the event without using the general addition rule.
The probability that the age of the person obtained is either between 25 and 59, inclusive, or at least 35 is $\square$
b. Find the probability in part (a) using the general addition rule. Identify the three probabilities used in the general addition rule.
The probability that the age of the person obtained is between 25 and 59, inclusive, is $\square$. The probability that the age of the person obtained is at least 35 is $\square$. The probability that the age of the person obtained is between 25 and 59, inclusive, and at least 35, is $\square$.
(Type integers or decimals rounded to three decimal places as needed)
Identify the probability of the event using the general addition rule.
Using the general addition rule, the probability that the age of the person obtained is either between 25 and 59, inclusive, or at least 35 is $\square$
(Type an integer or a decimal. Round to three decimal places as needed)
Answer (1)
Calculate the total number of people: 404.789 .
Calculate the number of people aged 25 to 59 inclusive or at least 35: 394.326 .
Calculate the probability without the general addition rule: 404.789 394.326 ≈ 0.974 .
Using the general addition rule: P ( 25 to 59 ) + P ( at least 35 ) − P ( 25 to 59 and at least 35 ) = 0.852 + 0.679 − 0.558 = 0.973 . The probability is 0.974 .
Explanation
Problem Analysis First, let's analyze the problem. We are given a table of age ranges and their corresponding frequencies. We need to find the probability that a person's age is either between 25 and 59 inclusive, or at least 35. We will solve this in two ways: first, without using the general addition rule, and then using the general addition rule.
Solving without General Addition Rule Without using the general addition rule, we need to determine the number of people whose age is between 25 and 59 inclusive OR at least 35, and divide that by the total number of people. This means we sum the frequencies from the 25-29 age range up to the 75 & over age range. This corresponds to option A.
Calculate Total Number of People The total number of people is the sum of all frequencies: 5.965 + 4.498 + 58.959 + 60.343 + 66.397 + 58.935 + 45.404 + 31.542 + 23.401 + 10.303 + 29.063 + 9.979 = 404.789 .
Calculate Number of People Aged 25 to 59 or At Least 35 The number of people with ages between 25 and 59 inclusive or at least 35 is the sum of frequencies from 25-29 to 75 & over: 58.959 + 60.343 + 66.397 + 58.935 + 45.404 + 31.542 + 23.401 + 10.303 + 29.063 + 9.979 = 394.326 .
Calculate Probability without General Addition Rule The probability without using the general addition rule is the number of people aged 25 to 59 or at least 35, divided by the total number of people: 404.789 394.326 ≈ 0.974 .
General Addition Rule Now, let's use the general addition rule. The general addition rule states that P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ) , where P ( A ∪ B ) is the probability of A or B, P ( A ) is the probability of A, P ( B ) is the probability of B, and P ( A ∩ B ) is the probability of A and B.
Probability of Age Between 25 and 59 We need to find the probability that the age of the person obtained is between 25 and 59 inclusive. This is the sum of frequencies from 25-29 to 55-59, divided by the total number of people: 404.789 58.959 + 60.343 + 66.397 + 58.935 + 45.404 + 31.542 + 23.401 = 404.789 344.981 ≈ 0.852 .
Probability of Age At Least 35 We need to find the probability that the age of the person obtained is at least 35. This is the sum of frequencies from 35-39 to 75 & over, divided by the total number of people: 404.789 66.397 + 58.935 + 45.404 + 31.542 + 23.401 + 10.303 + 29.063 + 9.979 = 404.789 275.024 ≈ 0.679 .
Probability of Age Between 25 and 59 AND At Least 35 We need to find the probability that the age of the person obtained is between 25 and 59 inclusive AND at least 35. This is the sum of frequencies from 35-39 to 55-59, divided by the total number of people: 404.789 66.397 + 58.935 + 45.404 + 31.542 + 23.401 = 404.789 225.679 ≈ 0.558 .
Calculate Probability Using General Addition Rule Using the general addition rule, the probability is 0.852 + 0.679 − 0.558 = 0.973 .
Final Answer Therefore, the probability that the age of the person obtained is either between 25 and 59 inclusive, or at least 35 is approximately 0.974.
Examples
This type of probability calculation is useful in market research. For example, a company might want to know the probability that a customer is either in a certain age range or has a certain income level to target their advertising effectively. By analyzing demographic data, they can determine these probabilities and optimize their marketing strategies. This ensures that the advertisements reach the intended audience, increasing the likelihood of sales and maximizing the return on investment.
