a. Without 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 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, 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 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 $\square$
(Type an integer or a decimal. Round to three decimal places as needed.)
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. Round 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 (2)
Calculate the total number of people: 449018 .
Calculate the number of people whose age is between 25 and 59 (inclusive) or at least 35: 394163 .
Calculate the probability without using the general addition rule: 449018 394163 ≈ 0.878 .
Calculate the probability using the general addition rule: 0.761 + 0.612 − 0.495 = 0.878 .
0.878
Explanation
Problem Analysis We are given a table of ages and their corresponding frequencies. We want to find the probability that a person's age is either between 25 and 59 (inclusive) or at least 35. We will first compute this probability without using the general addition rule, and then using the general addition rule.
Calculate total frequency for combined age ranges a. Without using the general addition rule, we need to find the number of people whose age is between 25 and 59 (inclusive) or at least 35. This means we need to consider the age ranges 25-29, 30-34, 35-39, 40-44, 45-49, 50-54, 55-59, 60-64, 65-74, and 75 & over. We sum the frequencies for these age ranges.
Sum the frequencies The sum of the frequencies for the age ranges 25-29, 30-34, 35-39, 40-44, 45-49, 50-54, 55-59, 60-64, 65-74, and 75 & over is: 56956 + 62343 + 66537 + 56635 + 44084 + 31542 + 23401 + 16893 + 29093 + 6679 = 394163
Calculate total number of people Next, we need to find the total number of people, which is the sum of all the frequencies: 5968 + 48887 + 56956 + 62343 + 66537 + 56635 + 44084 + 31542 + 23401 + 16893 + 29093 + 6679 = 449018
Calculate the probability The probability that the age of the person obtained is either between 25 and 59, inclusive, or at least 35 is the number of people whose age is between 25 and 59 (inclusive) or at least 35, divided by the total number of people: 449018 394163 ≈ 0.878
State the answer for part a Therefore, the probability that the age of the person obtained is either between 25 and 59, inclusive, or at least 35 is approximately 0.878.
The correct answer for how to compute the probability without using the general addition rule is 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.
Calculate P(25 to 59) b. Using the general addition rule, we need to find the probability that the age of the person obtained is between 25 and 59, inclusive, the probability that the age of the person obtained is at least 35, and the probability that the age of the person obtained is between 25 and 59, inclusive, and at least 35.
The probability that the age of the person obtained is between 25 and 59, inclusive, is: 449018 56956 + 62343 + 66537 + 56635 + 44084 + 31542 + 23401 = 449018 341498 ≈ 0.761
Calculate P(at least 35) The probability that the age of the person obtained is at least 35 is: 449018 66537 + 56635 + 44084 + 31542 + 23401 + 16893 + 29093 + 6679 = 449018 274864 ≈ 0.612
Calculate P(25 to 59 and at least 35) The probability that the age of the person obtained is between 25 and 59, inclusive, and at least 35 is: 449018 66537 + 56635 + 44084 + 31542 + 23401 = 449018 222200 ≈ 0.495
Apply 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: P ( A or B ) = P ( A ) + P ( B ) − P ( A and B ) P ( A or B ) = 0.761 + 0.612 − 0.495 = 0.878
State the answer for part b Therefore, 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 approximately 0.878.
Examples
Understanding probability is very useful in real life. For example, insurance companies use probabilities to assess risk and determine premiums. Similarly, in public health, probabilities are used to predict the spread of diseases and to evaluate the effectiveness of interventions. In finance, probabilities are used to model market behavior and to make investment decisions.
The probability that a person is either between 25 and 59, inclusive, or at least 35 can be calculated directly (approximately 0.878) or using the general addition rule, which will also yield the same result. In both methods, careful attention was paid to avoid double-counting ages 35-59. Overall, both approaches confirm the same findings about the population's age distribution.
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