STA113: Probability and Statistics in Engineering

Homework #1

Due: Friday, Jan 30

Problem 1

There are 20 people at a party. How likely is it that at least two of the 20 people have the same birthday? To simplify, ignore leap years, and assume that all days are equally likely to be birthday. Then:

a) Compute the above probability analytically using the rule for the equally likely simple events.

b) Estimate the probability in question using the limiting frequency idea. To do this you have to use simulation to generate a number of different parties, and then count how many the simulated parties include at least two people with the same birthday. In Minitab,

will generate 25 parties of 20 people each with equally likely birthdays. Each party is stored in a column. Then use

to display the results. How many of your 25 sets of 20 birthdays had no matching birthdays? What is the estimate of the probability of interest based on the simulation?

Problem 2

Beset by budget problems, both Engineering and Physics decide to buy routers (one each) from from Bubba's Bridge & Router Company. In Bubba's inventory there are nine routers, three of which are defective.

a) If Physics buys theirs first and it works, what is the probability that our (Engineering) router works?

b) If Physics buys theirs first and it fails, what is the probability that our (Engineering) router works?

c) Is it in our best interest to get Physics to buy theirs first? Why?

d) If we buy ours first, what is the chance that it works, given that Physics buys theirs second and it works? (note unexpected order)

Problem 3

In a piece of electronic equipment, a gate, which may be open or closed, has a random method of operation. Every millisecond it may change its state; the changes occur in the following manner:

  1. If it is open it remains open with probability 1-a and becomes closed with probability a;
  2. If it is closed it remains closed with probability 1-b and becomes open with probability b.

a) If 0<a<1 and 0<b<1, and Pn is the probability that the gate is closed after n milliseconds, derive a recurrence relation for Pn (i.e. express it in terms of Pn-1), showing clearly how your relation was obtained. Let P0 be the probability that the gate is closed initially. Solve your recurrence relation to give Pn in terms of a, b, P0, and n. Make sure that you justify your solution.

b) Find the limit as n->oo and explain intuitively why it doesn't depend on P0.

c) Find Pn in the four special cases a=0,b=0; a=0,b=1; a=1,b=0; and a=1,b=1. Explain the behaviour of the system in each case.

Problem 4

Problem 3.57, on page 138 of the textbook (Mendenhall & Sincich).

Optional Exercise

Show by example that there exist events A and B such that the conditional probability of A, given B, P[A|B], can be the same, or greater, or smaller than the probability P[A] of A. It is enough to give the probabilities of A, B, and their intersection in each case.

Good Luck!