A mutual fund company has 6 funds that invest in the U.S....

A mutual fund company has 6 funds that invest in the U.S. market and 4 that invest in international markets. A customer wants to invest in two U.S. funds and 2 international funds. a. How many different sets of funds from this company could the investor choose? b. Unknown to this investor, one of the U.S. funds and one of the international funds will seriously underperform next year. If the investor selects funds for purchase at random, what is the probability that at least one of the chosen funds will seriously underperform next year?

A mutual fund company has 6 funds that invest in the U.S. market and 4 that invest in international markets. A customer wants to invest in two U.S. funds and 2 international funds.
a. How many different sets of funds from this company could the investor choose?
b. Unknown to this investor, one of the U.S. funds and one of the international funds will seriously underperform next year. If the investor selects funds for purchase at random, what is the probability that at least one of the chosen funds will seriously underperform next year?

Key Concepts

Combination

A combination is a method of selecting items from a larger set without regard to the order in which they are selected. It is computed using the binomial coefficient (often read as 'n choose k') and is fundamental in counting the number of ways to choose subsets, such as selecting funds from a group.

Multiplication Principle

The multiplication principle is a fundamental counting technique stating that if one event can occur in a certain number of ways and a second independent event can occur in another number of ways, then the total number of ways the two events can occur together is the product of the individual counts. This principle allows the independent selection of funds from separate categories to be counted together.

Complement Rule

The complement rule in probability states that the probability of an event occurring is equal to one minus the probability of that event not occurring. This is particularly useful when calculating the probability of 'at least one' occurrence by subtracting the probability of 'none' occurring from one.

Probability in Combinatorial Settings

Using combinatorial techniques to determine both the total number of possible outcomes and the number of favorable outcomes connects counting with probability. This approach allows for the calculation of the likelihood of various events, such as selecting funds that may underperform, by leveraging the foundational principles of combinations and probability.

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