Mathematics 21a -- Computer Science Section

Week 10 homework assignment for computer section

  • This is the first installment of a single assignment on the material covered in class on Nov. 15 and Nov. 20.
  • The entire assignment is due on Tuesday, Nov. 27.
  • Of course, you are encouraged to work these problems before Nov. 20.

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".