Permutations and Combinations s1

CEE11 Practice Problems (stolen from various Probability textbooks)

Permutations and Combinations

1. How many 5-digit numbers can be formed from the integers 1, 2, 4, 6, 7 and 8 if no integer can be used more than once? How many of these numbers will be even?

2. In how many ways can 3 letters be mailed in 6 mailboxes if (a) each letter must be mailed in a different box? (b) letters are not necessarily mailed in different mailboxes? If the letters are mailed at random and not necessarily in different boxes, what is the probability that all the letters are put in the same mailbox?

3. There are 10 chairs in a row. In how many ways can 2 persons be seated? In how many of these ways will the 2 persons be sitting in adjacent chairs? In how many ways will they have at least one chair between them?

4. In how many ways can 5 red balls, 4 black balls and 4 white balls be placed in a row such that the balls at the ends of the row are of the same colour?

Probability

1. In the World Series, Teams A and B play until one team has won 4 games. Let p be the probability that team A wins any individual game played with B. Then q = 1 – p is the probability that Team B wins the game. Use this information to answer the following questions. (a) What is the probability that A wins in 4 games? That B wins in 4 games? (b) What is the probability that A wins the series in the 5th game? That the series ends in 5 games? 2. An ordinary deck of cards is shuffled and the cards are dealt face up one at a time, until an ace appears. What is the probability that the first ace appears at the 4th card? At the 47th card? (Speaking of cards and card games, remember Canada Bill Jone's Motto: It's morally wrong to allow suckers to keep their money. And its supplement: A .44 magnum beats four aces.)

3. A student is to answer 8 out of 10 questions on an exam. In how many ways can he do that? What is the probability that he has answered the first 3 questions? What is the probability that if he has answered at least 4 out of the first 5 questions?

4. The famous Lewis Carroll's Pillow Problem: A bag contains a counter, known to be either white or black. What is the chance of drawing a white counter? Now, a white counter is put in, the bag is shaken, and a counter is drawn out, which turns out to be white. What is the chance of drawing a white counter now?

Bayes Theorem

1. Assume that 1 coin in 10,000,000 has two heads; the rest are fair. If a coin, chosen at random, is tossed 10 times and comes up “heads” every time, what is the probability that it is two headed?