An auto insurance company has 12,000 policyholders. Each policyholder is classified as Young or Old, Male or Female, and Married or Single. Of the policyholders, 4,000 are young, 4,500 are male, and 7,500 are married. In addition, 1,400 are young and male, 3,100 are married and male, and 1,500 are young and married. Finally, 500 of the policyholders are young, married, and male. How many of the company’s policyholders are (b) Young or female, and single?
The correct answer and explanation is:
To solve this problem, we’ll use the principle of inclusion and exclusion to find the number of policyholders who are Young or Female and Single.
Given Data:
- Total policyholders: 12,000
- Young: 4,000
- Male: 4,500
- Female: 12,000 − 4,500 = 7,500
- Married: 7,500
- Single: 12,000 − 7,500 = 4,500
- Young ∩ Male: 1,400
- Married ∩ Male: 3,100
- Young ∩ Married: 1,500
- Young ∩ Married ∩ Male: 500
Step-by-step Solution:
We are asked for the number of (Young or Female) AND Single.
Let’s define sets:
- Let Y = Young = 4,000
- Let F = Female = 7,500
- Let S = Single = 4,500
We want to find:
(Y ∪ F) ∩ S
This is equal to:
[(Y ∩ S) ∪ (F ∩ S)]
But since Y and F can overlap, we use inclusion-exclusion:
(Y ∩ S) + (F ∩ S) − (Y ∩ F ∩ S)
Let’s find each part:
1. Y ∩ S:
We know:
- Y = 4,000
- Y ∩ Married = 1,500
So, Y ∩ Single = 4,000 − 1,500 = 2,500
2. F ∩ S:
We know:
- F = 7,500
- Married = 7,500
- Married ∩ Male = 3,100
So, Married ∩ Female = 7,500 − 3,100 = 4,400
Thus, Single ∩ Female = F − Married ∩ Female = 7,500 − 4,400 = 3,100
3. Y ∩ F ∩ S:
We already know:
- Y ∩ Married = 1,500
- Y ∩ Married ∩ Male = 500
So, Y ∩ Married ∩ Female = 1,500 − 500 = 1,000
Hence, Y ∩ Single ∩ Female = (Y ∩ Female) − (Y ∩ Married ∩ Female)
We don’t know Y ∩ Female directly, but we can find:
Y = 4,000
Y ∩ Male = 1,400
So, Y ∩ Female = 4,000 − 1,400 = 2,600
So, Y ∩ Single ∩ Female = 2,600 − 1,000 = 1,600
Now apply inclusion-exclusion:
(Y ∩ S) + (F ∩ S) − (Y ∩ F ∩ S)
= 2,500 + 3,100 − 1,600
= 4,000
✅ Final Answer:
4,000 policyholders are Young or Female, and Single.
Explanation (300 words):
This problem combines principles of set theory, especially inclusion and exclusion, to find the number of people satisfying a compound condition: those who are Young or Female, and Single. Since each policyholder is classified by three characteristics (age, gender, and marital status), we must carefully account for overlapping categories.
We began by identifying how many policyholders are Young and Single. From the total number of young policyholders (4,000) and the number who are also married (1,500), we deduced that the remaining 2,500 must be single.
Next, we considered the number of Female and Single policyholders. Given that there are 7,500 female policyholders and that 4,400 of them are married (deduced by subtracting male married policyholders from the total married), the number of single females is 3,100.
However, some individuals could be counted twice — those who are both young and female and single. To avoid overcounting, we determined this group using complementary information: of the 2,600 young females, 1,000 are married, so 1,600 must be single. We subtracted this overlap from the total.
Using the inclusion-exclusion formula:
(Young ∩ Single) + (Female ∩ Single) − (Young ∩ Female ∩ Single), we arrived at:
2,500 + 3,100 − 1,600 = 4,000.
This type of problem is common in statistics and probability, where accurate counting in overlapping sets is crucial.