- Thread starter valnadam
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Stocks ==> AND + Stocks Only = 34%

Bonds ==> AND + Bonds Only = 30%

Stocks or Bonds ==> Stocks + Bonds - AND (since both Stocks and Bonds include the AND area) = 35%

Abandoning the lengthy names:

A + S = 0.34 ==> S = 0.34 - A

A + B = 0.30 ==> B = 0.30 - A

S + B + A = 0.35

A couple of substitutions, a little algebra, and you are done.

In this case, the event A = event that person invests in stocks, and the event B = person invests in bonds

So P( A ) = 0.34 (decimal form of 34%), P( B ) = 0.30 and P(A or B) = 0.35

What we want to find is P(A and B), let's make x = P(A and B). Now let's use the following formula

P(A or B) = P( A ) + P( B ) - P(A and B)

Plug in the values described above:

0.35 = 0.34 + 0.30 - x

Now simplify and solve for x

0.35 = 0.34 + 0.30 - x

0.35 = 0.74 - x

0.35 - 0.74 = -x

-0.39 = -x

0.39 = x

x = 0.39

So because x = 0.39, this means that P(A and B) = 0.39

30% end up investing in bonds, and 35% end up investing in stocks or bonds (or both).

What is the probability that a person who inquires about investments at this firm will invest in both stocks and bonds?

Are you familiar with this formula?

. . \(\displaystyle P(A \vee B) \;=\; P(A) + P(B) - P(A \wedge B)\)

\(\displaystyle \text{We are given: }\(S) = 0.34,\;P(B) = 0.30,\;P(S \vee B) = 0.35\)

\(\displaystyle \text{Substitute into: }\;\underbrace{P(S \vee B)}_{0.35} \;=\;\underbrace{P(S)}_{0.34} + \underbrace{P(B)}_{0.30}\,-\,P(S \wedge B)\)

\(\displaystyle \text{We have: }\;0.35 \;=\;0.34 + 0.30 - P(S \wedge B)\)

. . . .\(\displaystyle P(S \wedge B) \;=\;0.34 + 0.30 - 0.35\)

. . . .\(\displaystyle P(S \wedge B) \;=\;0.29\)