Question

In a survey of 200 employees of a company regarding their 401(k) investments, the following data was obtained.

152 had investments in stock funds
83 had investments in bond funds
56 had investments in money market funds
51 had investments in stock and bond funds
34 had investments in stock funds and money market funds
35 had investments in bond funds and money market funds
23 had investments in stock funds, bond funds, and money market funds


(a) What is the probability that an employee of the company chosen at random had investments in exactly two kinds of investment funds?

Answer 1

(b) 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.)

Answer 2

(c) 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.)

Answer 3

Construct a Venn diagram. I'll let \(S,B,M\) denote stock, bond, and money market funds, respectively.
The notation should be clear: \(SB\) denotes the intersection of \(S\) and \(B\), etc.

Answer 4

23 of the 200 employees have investments in all three types of funds, so the corresponding probability is \(P(S\cap B\cap M)=\dfrac{23}{200}=0.115\).

Answer 5

51 had investments in \(S\) and \(B\), but we've already counted some of these employees in the group that had invested in all three funds. Using the counting principle, \(|A\cap B|=|A|+|B|-|A\cup B|\), we determine that \(51-23=28\) employees belong to \(SB\).

Similarly, you can find that \(BM\) and \(SM\) contain \(35-23=12\) and \(34-23=11\) employees, respectively. So, \(P(B\cap M)=\dfrac{12}{200}=0.06\) and \(P(S\cap M)=0.055\), respectively.

Answer 6

Thank you so much! I understand now!

Answer 7

yw!

Try extending this reasoning to determine the sizes of \(S,B,M\).