In a survey of 200 employees of a company regarding their 401(k) investments, the following data were obtained.
150 had investments in stock funds.
78 had investments in bond funds.
60 had investments in money market funds.
46 had investments in stock funds and bond funds.
38 had investments in stock funds and money market funds.
36 had investments in bond funds and money market funds.
24 had investments in stock funds, bond funds, and money market funds.
What is the probability that an employee of the company chosen at random had investments in exactly one kind of investment fund? (Enter your answer to two decimal places.)
What is the probability that an employee of the company chosen at random had no investment in any of the three types of funds? (Enter your answer to three decimal places.)
To find the probability that an employee had investments in exactly one kind of investment fund, we need to consider the number of employees who had investments in only one kind of fund.
Let's calculate the number of employees who had investments in only stock funds, bond funds, or money market funds:
Number of employees with investments in only stock funds = (Number of employees with investments in stock funds) - (Number of employees with investments in stock funds and bond funds) - (Number of employees with investments in stock funds and money market funds) + (Number of employees with investments in stock funds, bond funds, and money market funds)
= 150 - 46 - 38 + 24
= 90
Number of employees with investments in only bond funds = (Number of employees with investments in bond funds) - (Number of employees with investments in stock funds and bond funds) - (Number of employees with investments in bond funds and money market funds) + (Number of employees with investments in stock funds, bond funds, and money market funds)
= 78 - 46 - 36 + 24
= 20
Number of employees with investments in only money market funds = (Number of employees with investments in money market funds) - (Number of employees with investments in stock funds and money market funds) - (Number of employees with investments in bond funds and money market funds) + (Number of employees with investments in stock funds, bond funds, and money market funds)
= 60 - 38 - 36 + 24
= 10
Now, the probability that an employee had investments in exactly one kind of investment fund is the sum of probabilities of each of these cases divided by the total number of employees:
Total number of employees = 200
Probability = (Number of employees with investments in only stock funds + Number of employees with investments in only bond funds + Number of employees with investments in only money market funds) / Total number of employees
= (90 + 20 + 10) / 200
= 120 / 200
= 0.60
Therefore, the probability that an employee of the company chosen at random had investments in exactly one kind of investment fund is 0.60.
To calculate the probability that an employee had no investment in any of the three types of funds:
Number of employees with no investment in any of the three types of funds = Total number of employees - (Number of employees with investments in stock funds + Number of employees with investments in bond funds + Number of employees with investments in money market funds) + (Number of employees with investments in stock funds, bond funds, and money market funds)
= 200 - (150 + 78 + 60) + 24
= 36
Probability = Number of employees with no investment in any of the three types of funds / Total number of employees
= 36 / 200
= 0.180
Therefore, the probability that an employee of the company chosen at random had no investment in any of the three types of funds is 0.180.
To find the probability that an employee had investments in exactly one kind of investment fund, we need to calculate the number of employees with investments in only one type of fund and divide it by the total number of employees.
To do this, we can use the principle of inclusion-exclusion.
1. First, we calculate the total number of employees with investments in only stock funds, only bond funds, and only money market funds.
- Employees with only stock funds: 150 - 46 - 38 + 24 = 90
- Employees with only bond funds: 78 - 46 - 36 + 24 = 20
- Employees with only money market funds: 60 - 38 - 36 + 24 = 10
2. Next, we sum up the number of employees with investments in only one type of fund: 90 + 20 + 10 = 120.
3. Finally, we divide the number of employees with investments in only one type of fund by the total number of employees: 120 / 200 = 0.60.
Therefore, the probability that an employee chosen at random had investments in exactly one kind of investment fund is 0.60.
To find the probability that an employee had no investments in any of the three types of funds, we need to calculate the number of employees with no investments in any of the funds and divide it by the total number of employees.
To do this, we can subtract the number of employees with investments in at least one type of fund from the total number of employees.
1. First, we calculate the number of employees with investments in at least one type of fund.
- Employees with investments in at least one type of fund = Total employees - Employees with no investments in any fund
- Employees with no investments in any fund = Total employees - (Employees with investments in stock funds + Employees with investments in bond funds + Employees with investments in money market funds)
- Employees with no investments in any fund = 200 - (150 - 24 - 38 + 36 + 78 - 24 - 46 + 36 + 60 - 24 - 38 + 36)
- Employees with no investments in any fund = 200 - (150 - 24 - 38 + 36 + 78 - 24 - 46 + 36 + 60 - 24 - 38 + 36)
- Employees with no investments in any fund = 200 - 96
- Employees with no investments in any fund = 104
2. Finally, we divide the number of employees with no investments in any of the funds by the total number of employees: 104 / 200 = 0.52.
Therefore, the probability that an employee chosen at random had no investment in any of the three types of funds is 0.52.