Problems: Discrete Probability Distributions
Part 2
10. You are playing a game in which you spin a wheel as they do in Wheel of Fortune. The wheel can stop at four places:
|
x: |
$100 |
$50 |
$10 |
$0 |
|
p(x): |
.1 |
.2. |
.3 |
? |
b) What is the expected value of this game?
11. Pepsi had an i-Tunes promotion in which one in three bottles won a free song. If John bought a bottle of Pepsi from a machine three mornings in a row,
- a) what was the probability that all three bottles would be winners?
- b) what was the probability that all three would be losers?
- c) what was the probability that exactly one of the three would be a winner? (Hint: What is the probability of winning on the first day and losing on days two and three? How many other ways are there to get exactly one win?)
12. Are these probability distributions? Explain why or why not.
- a) P(x) = .5 - .1x for x = 1, 2, 3, 4
- b) P(a<x<b) = b-a for x between 0 and 1
- c) P(x) = (.5)x for x = 1, 2, 3, …..
13. P(X) = x/15 for x = 1, 2, 3, 4, 5 is a probability distribution.
- a)What is the expected value of this distribution?
- b) What is the probability of an odd number occurring?
- c) What is the probability that at least a 3 will occur?
- d) Use the answers from b and c plus the law of addition to find the probability that either an odd number or at least a 3 will occur.
14.Show that the following is or is not a valid
probability distribution:
For x = 0, 1, 2, 3: P(x) = x!/10
(x! is x factorial.)
15. For x = 2, 3, 4, let P(x) = (x*(x+1))/38.
Is this a probability distribution? Why or why not? If it is
a probability distribution, calculate its mean.
16. Suppose a bill can be selected from one of three hats. In the first hat there are five one-dollar bills and five ten-dollar bills. In the second hat there are three fives and one ten, and in the third there are four tens.
- a) A hat will be selected at random and then a bill will be drawn from it. What is the probability that, if a ten dollar bill is selected by this procedure, it will have come from the first hat?
- b) Suppose the hat is selected in this way: a die is tossed, and if a one, two, or three comes up, the first hat is selected. If a four or five comes up, the second hat is selected. If a six comes up, the third hat is selected. What is the probability that a person will select a one-dollar bill with this method?
- c) If, with the method in part b, a ten-dollar bill has been selected, what is the probability that it came from the third hat?
17. John has constructed a wheel like the wheel on "Wheel of Fortune." It has twenty slots on which the arrow can land, and the arrow is equally likely to land on any one of the twenty slots. He paints fifteen of the slots white, four green and one red. He wants to use this wheel as a gambling game to raise money for charity, and decides that he will pay nothing if it lands on the white, one dollar if it lands on green, and $20 if it lands on the red slot.
- a) What is the expected payout of this game?
x p(x) 0 15/20 $1 4/20 $20 ??? - b. John finally decides he wants to charge one dollar for this game. He wants to keep the payouts on white and green as they were in the previous question. What is the largest he can have on red and not expect to lose money in the long run?
x p(x) +$1 15/20 $0 4/20 $?? ??? - c. What is the standard deviation of the distribution you used in part b?
18. Suppose that you have a regular tetrahedron (a polyhedron with four surfaces) and a cube. The surfaces of the tetrahedron are numbered one through four, the cube one through six. These polyhedrons are tossed together, and the combinations that are on the bottom are recorded.
- a) Construct the sample space of all possible outcomes and assign probabilities to each event.
- b) What is the probability that both polyhedrons will hide a number two or less? (How do you get your answer?
- c) What is the probability that they will hide a total of seven? (How did you get your answer?)
19. Suppose that you have a regular tetrahedron (a polyhedron with four surfaces) and a cube. The surfaces of the tetrahedron are numbered one through four, the cube one through six. These polyhedrons are tossed together, and the combinations that are on the bottom are recorded.
- a) Construct the sample space of all possible outcomes and assign probabilities to each event.
- b) What is the probability that both polyhedrons will hide a number two or less? (How do you get your answer?
- c) What is the probability that they will hide a total of seven? (How did you get your answer?)
20. What is the probability that four cars drawn at random from a deck of cards will all be red? (Assume that they are not replaced.)
21. Consider the following rule for assigning probabilities:
|
Event: |
3 |
4 |
5 |
|
Probability: |
2/3 |
1/6 |
1/6 |
a) Does this rule fulfill the requirements to be a probability distribution? Explain why or why not.
b) What is the probability that either a 4 or a 6 will occur if we take one observation from this distribution?
c) If we took a very large sample from this distribution, say 1000 observations, and found the sample mean, what number should this sample mean be very close to?
22. Suppose you take a die and alter it, changing the five into a two. The sample space now has these possibilities: 1, 2, 3, 4, 6. What are the probabilities that should be attached to each? What is the expected value or mean of this probability distribution?
(In my first years teaching statistics, I always spent time on the binomial distribution. Over time I came to believe that the time used to discuss it could be better used for other things, so I dropped it. I never did use it to help explain the Central Limit Theorem, which might be its best use. When n gets large, the binomial distribution gets very close to the Normal distribution.
For some binomial distribution questions, go here.)