Consider a company that has a retirement plan for its employees in which the money contributed for each employee can be divided (at the employee's choice) between stocks, bonds and certificates of deposit (CDs) An employee may choose to put all of his or her money into one investment (e.g., stocks) or may divide it between the three (say, one-third into each, or half each into stocks and bonds with nothing into CDs). You have the following information regarding employee's decisions: 60% have at least some money in CDs 12% have all their money in CDs 10% have all their money divided between stocks and CDs 16% have all their money divided between bonds and CDs 20% have all their money in stocks 66% have atleast some money in stocks (a) What percent of employees have money in all three? (b) What percent of employees have all their money divided between stocks and bonds? (c) What percent of employees have all their money in bonds? (d) What percent of employees have atleast some money in bonds?

P(C_total)=0.60 as 60% of all employees have at least some money in Certificate Deposits
P(AC)=0.10 as 10% of employees have their money divided in Stocks and Certificate Deposits.

P(BC)=0.16 as 16% of employees have their money divided in Bonds and Certificate Deposits.

P(C)=0.12 as 12% of employees have all their money in Certificate Deposits.

P(A)=0.20 as 20% of employees have all their money in Stocks.

Now creating a venn diagram as attached in the figure gives following.

Part a

P(A∩B∩C) is the probability of employees having money in all three

Now As it is seen from the venn diagram

[tex]P(A\bigcap B\bigcap C) +P(AC)+P(BC)+P(C)=P(C_{total})\\P(A\bigcap B\bigcap C) +0.10+0.16+0.12=0.60\\P(A\bigcap B\bigcap C) =0.60-0.10-0.16-0.12\\P(A\bigcap B\bigcap C) =0.22[/tex]

So 22% of employees have money in all three.

Part b

P(AB) is the percentage of people with shared money in stocks and bonds

Now

[tex]P(A\bigcap B\bigcap C) +P(AC)+P(AB)+P(A)=P(A_{total})\\0.22 +0.10+P(AB)+0.20=0.66\\P(AB)=0.66-0.22 -0.10-0.20\\P(AB)=0.14[/tex]

So 14% of employees have their money shared in stocks and bonds.

Part d

P(B_total) is the percentage of people who have some of their money in the Bonds so

[tex]P(B_{total})=1-P(A)-P(C)-P(AC)\\P(B_{total})=1-0.2-0.12-0.1\\P(B_{total})=0.58\\[/tex]

So 58% people have at-least some their money in bonds.

Part c

P(B) is the percentage of people with all their money in bonds Now

So

[tex]P(A\bigcap B\bigcap C) +P(BC)+P(AB)+P(B)=P(B_{total})\\0.22 +0.16+P(B)+0.14=0.58\\P(B)=0.58-0.22 -0.16-0.14\\P(B)=0.06[/tex]

So 6% of employee have all their money in bonds.