A mutual fund company has six funds that invest in the U.S. market and four funds that invest in foreign markets. An investor wishes to create a mutual fund portfolio composed of two U.S. funds and two foreign funds. Unknown to this investor, one of the U.S. funds and one of the foreign funds will seriously under-perform next year.
If the investor selects funds for his portfolio at random, what is the probability that he will have selected a portfolio with at least one under-performing fund in it?
Probability of choosing ONLY good funds from the US market
= (5/6)*(4/5) = 2/3
Probability of choosing ONLY good funds from the foreign market
= (3/4)*(2/3)=2/4=1/2
Probability of choosing ONLY good funds from both markets = (2/3)*(1/2)=1/3
Probability of choosing at least one bad fund
= 1- (1/3) = 2/3
To find the probability of selecting a portfolio with at least one underperforming fund, we need to calculate the probability of NOT selecting a portfolio with all the funds performing well.
Let's break down the problem step by step:
Step 1: Calculate the probability of selecting two U.S. funds that perform well.
Since there are six U.S. funds, the probability of selecting a U.S. fund that performs well is 5/6. Since we need to select two U.S. funds that perform well, we multiply this probability by itself:
(5/6) * (5/6) = 25/36.
Step 2: Calculate the probability of selecting two foreign funds that perform well.
Since there are four foreign funds, the probability of selecting a foreign fund that performs well is 3/4. Similar to Step 1, we multiply this probability by itself:
(3/4) * (3/4) = 9/16.
Step 3: Calculate the probability of NOT selecting a portfolio with all funds performing well.
To find the probability of not selecting a portfolio with all funds performing well, we subtract the probability of selecting a portfolio with all funds performing well from 1.
The probability of selecting a portfolio with all funds performing well is:
(25/36) * (9/16) = 225/576.
Thus, the probability of not selecting a portfolio with all funds performing well is:
1 - 225/576 = 351/576.
Therefore, the probability of selecting a portfolio with at least one underperforming fund is 351/576.