Probability

There are 5 rotten plums in a crate of 25 plums. How many samples of 4 of the 25 plums contain: (a) only good plums? (b) Three good plums and 1 rotten plum? (c) One or more rotten plums?

Recursive relationship.

Consider the recursive relationship for combinations: C(n,r) = C(n-1,r) + C(n-1,r-1) • Prove this relationship algebraically using the mathematical definition of a combination, as well as that of the factorial function. • Provide a logical explanation for this relationship (Hint: consider n objects as consisting of n-1 existing objects plus a new nth object. Given the ultimate goal of selecting r objects from the entire set of n, you must select objects from the set of n-1 objects and decide how the new nth object should join the existing selected sets.)

Probability

Let E and F be non-zero-probability events. If E and F are mutually-exclusive, can they also be independent? Explain the answer, and also prove it algebraically using the definitions of mutually-exclusive and independent events.

Interest

An alumnus wants to set up a trust (which earns 9% interest compounded semi-annually) to provide a grant to his alma mater of $10,000 every six months, beginning six months from now, for six years. How much should be deposited in the trust?

investment

A corporation may obtain a machine by leasing it for six years (the useful life), paying an annual fee of $3,000, or by purchasing it for $12,000. Which alternative is preferable if the corporation can invest money at 8% per annum (i.e., its cost of capital is 8%)?