Mathematical Probability Additional Problems
Math 341, Fall 1999, Central College, Tom Linton

1. In a horse race, a finish is defined as an ordered triple consisting of the number (or name) of the winning horse, the number (or name) of the second place horse and the number (or name) of the third place horse, in that order. The finish 3,5,1 means horse number 3 won the race, horse number 5 came in second and horse number 1 was third. The finish 3,1,5 is different than the finish 3,5,1. Here are five types of wagers that gamblers can make at the track.
• Quinella This is a two-horse bet. The two horses you select must finish first and second, but it does NOT matter which horse is first and which comes in second. A 2-5 quinella is identical to a 5-2 quinella; both pay whenever horses 2 and 5 are the first two horses to cross the finish line (either 5 then 2, OR 2 then 5).
• Exacta This is a two horse bet in which the person making the wager must predict the winning horse and the second place horse. A 3-5 exacta will pay whenever horse 3 wins and horse 5 comes in second. A 5-3 exacta requires horse 5 to win and horse 3 to take second place. An exacta is like an ordered quinella.
• Perfecta This is a three horse bet similar to the quinella. One attempts to predict the top three horses in a race. It makes no difference which of the 3 selected horses wins, which selected horse comes in second and which selected horse takes third, but the 3 horses named in the bet must each finish in the top three places. A perfecta on horses 1,3 and 5 is the same as a perfecta on horses 3, 5 and 1 etc.
• Trifecta This is a three horse bet which can be described as an ordered perfecta. The bettor must accurately predict the winning horse, the second place horse and the third place horse. A 4-2-6 trifecta will pay only when horse 4 wins, horse 2 takes second and horse 6 finishes third.
• Place This is a single horse bet. The horse selected in the bet must finish first or second for the bet to pay.
Assume that there are nine horses in a race and each finish is equally likely.
1. How many finishes are there?
2. How many distinct quinella bets are possible?
3. How many distinct exacta bets are there?
4. How many distinct bets are there of type:
1. perfecta?
2. trifecta?
3. place?
5. How many finishes are covered by a single quinella? a single exacta? a single perfecta? a single trifecta? a single place? A finish is covered by a bet if the bet pays when that finish occurs.
6. If I purchase a 2-4 quinella, a place bet on horse 5 and a 1,2,6 perfecta, what percentage of all finishes have I covered?
7. Assuming horse 2 finished first, what is the likelihood that a 2-5 quinella paid off?
2. The probability function p(y) for a discrete RV Y is shown in the table below.

y	0	1	2	3	4	5
p(y)	0.22	0.41	0.27	0.08	0.015	0.005

1. What is P[Y = 2]? How about P[Y <= 2] (<= means less than or equal to)?
2. What is E[Y]?
3. What is the conditional probability that Y is even, given Y is greater than or equal to 4?
4. What is the conditional probability that Y is greater than or equal to four, given Y is odd?
3. Inspector Clouseau has compiled several eye-witness reports on the license plate of a pink car seen leaving the scene of a recent jewelry heist. The license consisted of 3 digits (0 to 9) followed by 4 letters (A to Z).
• The 4 letters were C, C, H and K (not necessarily in this order).
• The first number was a 1 or a 7.
• The second number was bigger than the first number.
• The third number was a 3 or an 8.
How many license plates does Clouseau have to check on?
4. An all male club is considering opening its membership to females. There are 30 current members of which 19 approve and 11 disapprove of opening the membership to females. A committee of 6 randomly chosen members will make the decision. Assume no member changes their current opinion.
1. How many committees are possible?
2. What is the probability that at least 3 members on the committee will favor the opening of membership?
3. What is the probability that at least four will be opposed?
5. A restaurant has 3 cooks, A, B and C. Each bakes their own variety of cake and with respective probabilities 0.02, 0.03 and 0.05, the cakes  made by cooks A, B and C will fail to rise. A bakes 50% of the cakes, B bakes 30% and C bakes 20%.
1. What proportion of failures are caused by A?
2. If a cake failed to rise, what is the probability that it was baked by B?
6. Pete's Pizza offers four sizes of pizzas, small, medium, large and the BFP. A pizza can be plain cheese or include any number of the seven additional toppings (sausage, pepperoni, canadian bacon, onions, green peppers, black olives or pineapple). How many types of pizza are there which are:
1. medium in size and have exactly 2 additional toppings?
2. not large and have exactly 3 additional toppings?
3. large or BFP and have 2 or 3 additional toppings.
7. From a well shuffled deck, you select a card, note what card it is, replace the card in the deck, re-shuffle and select another card, replace it, re-shuffle, etc., until you have seen and recorded 10 cards. Of the 10 selected and replaced cards, what is the probability that:
1. exactly 2 were clubs?
2. at least 5 were red?
3. you selected from 2 to 4 face cards?
4. you saw at least one ace?
8. This problem has a small enough sample space to enumerate. You can answer the questions by "counting" the points in the sample space which satisfy the given properties, OR by using more sophisticated techniques. Doing both will allow you to check your skills at using the more advanced methods. Basket A contains one black ball (B1) and 3 red balls (R1, R2 and R3). Basket B contains 2 black balls (B2, B3) and 2 red balls (R4, R5). In an experiment, one of the two baskets is chosen at random; then a first ball is drawn from that basket; then, without replacing the first ball, a second ball is drawn from the same basket. What are the probabilities that:
1. 2 red balls were selected? How about 2 black balls?
2. the 2 selected balls were the same color? the 2 balls were different colors?
3. the first ball was red? the second ball was black?
4. the balls were the same color, given that basket A was used? what if basket B was used?
5. the balls were different colors, given the first was red?
6. the balls were the same color, given the first was red?
7. the first ball was red, given the 2 balls were the same color?