CMSC 203 / 0201 Fall 2002

1. CMSC 203 / 0201Fall 2002 Week #9 – 21/23/25 October 2002 Prof. Marie desJardins

2. TOPICS • Permutations • Combinations • Binomial theorem • Discrete probability • Probability theory

3. MON 10/21PERMUTATIONS AND COMBINATIONS (4.3)

4. Concepts/Vocabulary • Permutation, r-permutation • P(n, r) = n! / (n-r)! • r-combination • C(n, r) = (n choose r) = n! / (r! (n-r)!) • Pascal’s identity • (n+1 choose k) = (n choose k-1) + (n choose k) • Pascal’s triangle • Binomial theorem • (x+y)n = j=0n (n choose j) xn-j yj

5. Examples • Exercise 4.3.1: List all the permutations of {a,b,c}. • Exercise 4.3.2: How many permutations are there of the set {a,b,c,d,e,f,g}? • How many permutations of a set of size k? • Exercise 4.3.3: How many permutations of {a,b,c,d,e,f,g} end with a?

6. Examples II • Exercise 4.3.19: A club has 25 members. • (a) How many ways are there to choose four members of the club to serve on an executive committee? • HINT: Which individual is in each of the four positions doesn’t matter • (b) How many ways are there to choose a president, vice president, secretary, and treasurer of the club? • HINT: Which individual is in each of the four positions does matter • Proof of binomial theorem (page 256)

7. WED 10/23DISCRETE PROBABILITY (4.4) ** HOMEWORK #6 DUE ** ** UNGRADED QUIZ TODAY!**

8. Concepts / Vocabulary • Experiment, sample space, event • Laplace’s probability – p(E) = |E| / |S| • OK for finitely many equally likely outcomes • p(~E) = 1 – P(E) • p(E1  E2) = p(E1) + p(E2) when E1, E2 are disjoint

9. Examples • Exercise 4.4.31: A roulette wheel has 38 numbers (18 red, 18 black, and 0 and 00, which are neither black or red). The probability that when the wheel is spun it lands on any particular number is 1/38. What is the probability that the wheel … • (a) …lands on a red number? • (b) … lands on a black number twice in a row? • (c) … lands on 0 or 00? • (d) … does not land on 0 or 00 five times in a row? • (e) … lands on a number between 1 and 6, inclusive, on one spin, but does not land between them on the next spin?

10. Examples II • Example 4.4.10 (an apparent paradox): You choose what’s behind Door #1. Before showing you what’s there, the host tells you that Door #2 is a losing door. Should you stick with Door #1 or switch with Door #3? • A door is a door is a door, isn’t it…? Shouldn’t the probability that the prize is behind Door #1 and the probability that the prize is behind Door #3 both be ½, since there are two doors left? (Exercise 4.4.35) • Why does telling you something about Door #2 tell you anything about Door #3??

11. Examples III • Exercise 4.4.34: Two events E1 and E2 are called independent if p(E1E2) = p(E1) p(E2). If a coin is tossed three times, which of the following pairs of events are independent: • (a) E1: the first coin comes up tails; E2: the second coin comes up heads. • (b) E1: the first coin comes up tails; E2: two, but not three, heads come up in a row. • (c) E1: the second coin comes up tails; E2: two, and not three, heads come up in a row.

12. FRI 10/11 - MON 10/14PROBABILITY THEORY (4.5)

13. Concepts and Vocabulary • Axioms of probability: for a set of mutually exclusive outcomes sS, • 0  p(s)  1 • sS p(s) = 1 • Event: set of outcomes • Conditional probability p(E|F) = p(EF) / p(F) • Independence p(EF) = p(E) p(F), or p(E|F) = p(E) • Bernoulli trials (2 outcomes) • Binomial distribution b(k:n, p) = (n choose k) pk qn-k • Random variables, expected values • Independent random variables, variance

14. Examples • Dice rolling: • Find the probability of each outcome when a biased die is rolled, if rolling a 2 or 4 are each three times as likely as rolling each of the other four numbers on the die. • What is the probability of rolling a 7 with two ordinary dice? • Exercise 4.5.5: Suppose a pair of dice is loaded. The probability that a 4 appears on the first die is 2/7 (other outcomes are 1/7), and the probability that a 3 appears on the second die is 2/7 (other outcomes are 1/7). What is the probability of rolling a 7 with these two dice?

15. Examples II • Independence • Exercise 4.5.10: Show that if E and F are independent events, then ~E and ~F are also independent events. • Exercise 4.5.11: If E and F are independent events, prove or disprove that ~E and F are necessarily independent events. • Conditional probability • Exercise 4.5.15: What is the conditional probability that exactly four heads apppear when a fair coin is flipped five times, given that the first flip came up heads? • Exercise 4.5.17: What is the c.p. that a random bit string of length 4 contains at lest 2 consecutive 0s, given that the first bit is a 1?

16. Examples III • Bernoulli trials: Exercise 4.5.26: Find each of the following probabilities when n independent Bernoulli trials are carried out with probability of success p. • (a) the probability of no successes • (b) the probability of at least one success • (c) the probability of at most one success • (d) the probability of at least two successes

17. Examples IV • Random variables and expected values: • Exercise 4.5.31: What is the expected sum of the numbers that appear on two dice, each biased so that a 3 comes up twice as often as each other number? • Exercise 4.5.32: What is the expected value of a $1 lottery ticket when the purchaser wins $10,000,000 iff the ticket contains the six winning numbers chosen from the set {1,2,3,…,50} (and nothing otherwise)? • Exercise 4.5.33: The 203 final exam contains 50 T/F questions (2 points each) and 25 multiple-choice questions (4 points each). • Linda answers a T/F question correctly with probability .9, and a multiple-choice question with probability .8. What is her expected score on the final? • What is the expected score of Emily, who hasn’t studied at all and answers T/F questions correctly with probability .5, and multiple-choice questions correctly with probability .25?