Please explain the following probability problem: in a survey of 200 employees of a company regarding their 401(k) investments, the following data were obtained.  141 had investments in stock funds. 89 had investments in bond funds. 56 had investments in money market funds. 48 had investments in stock funds and bond funds. 35 had investments in stock funds and money market funds. 33 had investments in bond funds and money market funds. 22 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? Enter your answer to three decimal places. (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.(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.

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There are 200 employees and four types of funds: stocks, bonds, money markets, and other.

141 employees invest in stocks.89 employees invest in bonds.56 employees invest in money markets.

These events are not mutually exclusive.

48 employees have stocks and bonds.35 employees have stocks and money markets. ...

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There are 200 employees and four types of funds: stocks, bonds, money markets, and other.

141 employees invest in stocks.
89 employees invest in bonds.
56 employees invest in money markets.

These events are not mutually exclusive.

48 employees have stocks and bonds.
35 employees have stocks and money markets.
33 employees have bonds and money markets.

22 employees invest in all three types of funds.

One approach to a problem like this is to create a Venn diagram. The rectangle encloses all 200 employees and the three intersecting circles represent the investment types. Anyone outside the three circles invests in some other way. (See attachment.)

Starting with the intersection of all three circles we have 22.

The intersection of stocks S and bonds B has 48 members, but we already have 22, so we put a 26 there. The intersection of S and money markets M has 35 members; with 22 accounted for, we put 13. In the intersection of B and M we have 33 members; subtracting the 22 we have 11.

There are 141 members in S, but we already have 26+22+13=61, so there are 60 members in S alone.
There are 89 members in B; we already have 26+22+11=59 so there are 30 members in B alone.
There are 56 members in M; we already have 13+22+11=46 so there are 10 members in M alone.

(a) The probability that a randomly chosen employee invested in exactly two funds is `26/200+11/200+13/200=50/200=0.25`

(b) The probability that an employee invested in exactly one kind of investment is `80/200+30/200+10/200=120/200=0.6`

(c) The probability that a randomly chosen employee invested in none of the three investments is `8/200=0.04`

Note that the probability of investing in all three is `22/200=0.11` and that .25+.6+.04+.11=1.

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