Stat 432: Homework 2
1. An urn contains 10 balls numbered 0, 1, 2, … , 9. In order to select a ball from the
urn you must pay $5. If you select a ball from the urn you win $1 times the number
on the ball (e.g. if 7 is selected you win $7). Let X equal the amount of money you
win on one selection from the urn.
a. Give the probability distribution for X.
b. What is the probability you win $5 or more?
c. Based on your answer in b) is this a fair game? Explain briefly.
d. What is the expected amount you win?
e. Based on your answer in d) is this a fair game? Explain briefly.
f. Suppose you draw from the urn twice, with replacement. Let Y be the total
amount you win. Give the probability distribution for Y.
g. What is the expected value of Y?
2. A shipment of 100 parts is delivered to a customer. Ten of the parts are defective
while ninety are good (non-defective). Parts are selected at random, with
replacement, and checked to see if the parts are defective or good.
a. What is the probability that 10 selected parts will have 1 defective? 2 defectives?
b. What is the probability that the first defective part is found on the 5th part
selected?
c. What is the probability that the second defective part is found on the 10th part
selected?
3. A shipment of 100 parts is delivered to a customer. Ten of the parts are defective
while ninety are good (non-defective). Parts are selected at random, without
replacement, and checked to see if the parts are defective or good.
a. What is the probability that 10 selected parts will have 1 defective? 2 defectives?
b. What is the probability that the first defective part is found on the 5th part
selected?
c. What is the probability that the second defective part is found on the 10th part
selected?
4. The average number of goals scored by both teams during the 90 minutes of a World
Cup Football (Soccer in the U.S.) match is 3. The number of goals scored by both
teams during a 90 minute World Cup Football match can be modeled as a Poisson
random variable.
a. What is the probability that at the end of 90 minutes a World Cup Football match
will be tied 0–0?
b. What is the probability that at the end of 90 minutes the score will be 1–0?
c. What is the expected number of goals scored during 90 minutes of a World Cup
Football match?
d. Based on the expected number of goals in c) how many goals will each team be
expected to have at the end of a 90 minute World Cup Football match?
e. For five independent World Cup Football matches what is the probability two will
end up with 1–0 scores at the end of 90 minutes?
f. Extra Credit: What is the chance a World Cup Football match will be tied at the
end of 90 minutes?