Probability

New member
Suppose that 34% of the people who inquire about investments at a certain brokerage firm end up investing in stocks, 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?

tkhunny

Moderator
Staff member
The "AND" are is shared. This is the key.

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.

New member
I got
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

tkhunny

Moderator
Staff member
Are you sure you want to go with that? The intersecton area is greater than either original area? I think not.

Make sure it makes sense before you are satisfied with it.

First, 0.34 + 0.30 = 0.64

soroban

Elite Member

Suppose that 34% of the people who inquire about investments at a certain brokerage firm end up investing in stocks,
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$$

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

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

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

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