1. Apostol, page 509, problem 12. This is the proof that Apostol
omits on page 494.
2. Section 13.18, page 500, problem 2.
3. You have to deliver crucial supplies using airplanes with very unreliable
engines. Each engine has a probability p of lasting for the entire
flight, and engine failures are independent events. If half
or more of the engines fail, the plane crashes. Your choice is between
using two-engine planes, which crash if either engine fails, or 4-engine
planes, which crash if two or more engines fail.
a. What is the probability that exactly three engines on a four-engine
plane will survive?
b. Determine for what value of p the probability that a plane will
not crash is the same for 2-engine and 4-engine planes.
c. For this value of p, would 6-engine planes be a better choice?
4. Exercises 3,4, 5, and 6 on page 505. These fill in some of
the proofs that Apostol omits on page 502.
5. Section 13.22, exercises 2 and 3. The answer to 3b) is 7/8.
6. Page 508, exercise 4. Apostol's answer to c) is 13/165.
7. Page 508, exercise 5. This is an example of what is called "Bayesian
inference," but from Apostol's viewpoint it is just another problem involving
conditional probabilities. Event A is "he selected the 2-headed coin"
and event B is "6 heads occurred".