CS 4953 Experimentation in Computer Science Assignment 2: Spring 2003

Due Thursday, February 27, 2003

You may use any method you like to do the following problems, but you must do them without asking other students. In each case indicate how you got your answer. (I looked it up is not sufficient.) Three significant figures is sufficient for numerical answers.

Note: Some of these problems are very easy. Some may be harder.

1. The game of Craps is played with two dice. In the simplest version you throw the dice and if you make a 7 or 11 you win. If you throw any other number, called the point you keep throwing the dice until you get the point again (and you win) or you get a 7 (and you lose). Find the probability of winning at Craps after you throw a 10.
2. Find the probability of winning at Craps.
3. What is the probability of being dealt a 5-card poker hand consisting of a royal flush?
   Optional: What is the probability of getting a royal flush in draw poker?
4. In bridge, each of 4 players is dealt 13 cards from a standard deck.
   What is the probability of getting all of the cards in one suit?
   What is the probability that you and your partner have all of the spades?
5. You have 10 black socks and 8 blue socks in a drawer. You pull out two socks at random. What is the probability that the two socks will match?
6. You have n black socks and m blue socks in a drawer. Both n and m are greater than or equal to 2. You pull out two socks at random. For what values of n and m is the probability of a match a minimum? What is that probability?
7. The bit error rate of a serial channel is 1 error in 100,000. Assuming independence of bit errors, what is the probability that a 1K block will be error free? What is the probability that a 1M block will be error free?
8. Suppose the bit error rate is p. In order to improve the error rate, each bit is sent n times, where n is an odd number. When n bits are received, the bit value with the most occurrences is used. Find the probability of error in terms of n. (You might want to use that n = 2k+1 for some k, and do the problem in terms of k.) You might want to start by finding the answer for k=1, 2, and 3.
9. The uniform distribution has a pdf that is constant over a closed interval, [a, b] and is zero outside the interval. Find the mean and standard deviation in terms of a and b.
10. The exponential distribution has a pdf that has the form c e^-ax for x > 0.
    a) Find the relationship between c and a.
    b) Find the a in terms of the mean of the distribution.
    c) Write a formula for the the pdf using only the mean (not a or c).
    d) Find the standard deviation in terms of the mean.
11. Suppose that x is N(0,1). Find the probability that x > 2.
12. Suppose that x is N(0,1). Find the value of y such that P(x<y) = .34.
13. Suppose that x is N(1,3). Find the probability that x > 2.
14. Suppose that x is N(-4,7). Find the value of y that makes P(-5 < x < y) = .34.
15. Suppose that x is N(0,1). Find the probability that x > 10.