Math 2000 - Counting and Probability Review Problems (Optional!)

Fall 2000

In the last two weeks we have studied several different methods for counting and several different methods for computing probabilities. The key to solving problems of this sort is to identify the appropriate method in each case. The following problems require the use of all of the methods we have studied. In each case identify the method(s) required and then solve the problem. Please show all of your work! Here are the methods we have studied.

Counting	Probability
Multiplication principle Arrangements with repetition Permutations Combinations	AND problems *Independent events *Dependent events OR problems *Mutually exclusive events *Non-mutually exclusive events At least once problems

1. A state lottery involves drawing 5 numbered balls from a barrel of 25 balls. How many different 5-number lottery tickets are possible?

2. In the lottery of problem 1, if you buy 10 tickets (with different numbers), what is the probability that exactly one of them is a winner?

3. The probability of randomly encountering a citizen of an Africa country on campus is 0.02. What is the probability that of the first five people you encounter on campus, at least one is a citizen of an Africa country?

4. Suppose you are dealt 4 cards from a standard deck of cards. What is the probability of being dealt four aces?

5. The 12 soccer teams of Felicity County are ranked each week according to the top 5 teams. How many top-5 rankings are possible?

6. Suppose you select and eat 3 candies, one at a time, from a bag that holds 15 caramels and 20 chocolates. What is the probability that you select two caramels and one chocolate?

7. You have just been assigned a random 3-digit password made up of the numerals 0, 1, 2,...,9. What is the probability that your password starts with 3?

8. Of the 50 men at a conference, 20 are Democrats and 30 are Republicans. Of the 70 women at a conference, 40 are Democrats and 30 are Republicans. What is the probability that the first person you meet at random is a Republican or a woman?

9. At the conference of problem 8, what is the probability that at least one of the first 3 people you meet (randomly) will be a man?

10. A ski swap offers 10 different kinds of skis, 5 different kinds of bindings, and 8 different kinds of boots. How many different ski packages are possible?

11. Suppose you are planning your 4-child family (ahead of time)! What is the probability that you will have two girls and then two boys in the order GGBB?

12. Referring to problem 11, what is the probability that you will have two girls and two boys in any order?

Answers (not complete solutions): 1. ₂₅C₅ = 53,130. 2. 1/5313 = 0.00019. 3. 0.096. 4. 0.0000037. 5. ₁₂P₅ = 95,040. 6. 0.11. 7. 0.01. 8. 100/120 = 0.83. 9. 0.80. 10. 400. 11. 1/16 = 0.0625. 12. 6/16 = 0.0375.