Download Additional Problems (Ch. 2)

STA 4321
Mathematical Statistics I
Additional Problems (Ch. 2)
Instructions: These problems will not be collected, so you are free to discuss them and work
together. Some of them may also be done as examples in class.
1. (Feller, An Introduction to Probability Theory and Its Applications, Vol I) Each of the 50 states has
two senators. A committee of 50 senators is to be selected at random. Let
A = {Florida is represented on the committee}
B = {all states are represented on the committee}
Find P (A) and P (B).
2. (Feller, An Introduction to Probability Theory and Its Applications, Vol I) At a bridge table, the 52
cards are randomly divided into four hands of 13 cards each. What is the probability that each of
the four hands contains an ace?
3. Suppose that a test for an performance enhancing drug has a “false positive rate” of 4%, and a
“false negative rate” of 3%, and that 1% of wiffleball players use the drug. A randomly selected
player tests positive for the drug. What is the probability that the player uses the drug? How does
your answer change if 10% of wiffleball players use the drug?
4. (Feller, An Introduction to Probability Theory and Its Applications, Vol I) It is known that 5 men
out of 100, and 25 women out of 10,000 are colorblind. A person is chosen at random and found to
be colorblind. What is the (conditional) probability that the person is a male? Assume that males
and females occur in equal numbers.
5. (Similar to exercise 2.137 in the 7th edition of Wackerly, Mendenhall, and Scheaffer, Mathematical
Statistics and Its Applications.) Ten urns each contain 10 balls. The kth urn contains k black and
10 − k white balls. An urn is selected at random and a ball is selected at random from that urn.
Given that the ball selected is black, what is the probability that it was taken from the kth urn?
6. (Exercises 2.119 and 2.138 in the 7th edition of Wackerly, Mendenhall, and Scheaffer, Mathematical
Statistics and Its Applications.) Suppose that two balanced dice are tossed repeatedly and the sum
of the two uppermost faces is determined on each toss.
(a) What is the probability that we obtain a sum of 4 before we obtain a sum of 7?
(b) Following is a description of the game of craps. A player rolls two dice and computes the total
of the spots showing. If the player’s first toss is a 7 or an 11, the player wins the game. If the
first toss is a 2, 3, or 12, the player loses the game. If the player rolls anything else (4, 5, 6, 8,
9 or 10) on the first toss, that value becomes the player’s point. If the player does not win or
lose on the first toss, he tosses the dice repeatedly until he obtains either his point or a 7. He
wins if he tosses his point before tossing a 7 and loses if he tosses a 7 before his point. What
is the probability that the player wins a game of craps?
7. (Feller, An Introduction to Probability Theory and Its Applications, Vol I) Die A has four red and
two white faces, whereas die B has two red and four white faces. A coin is flipped once. If it falls
heads, the game continues by throwing die A alone; if it falls tails, die B is to be used.
(a) Show that the probability of red at any throw is 1/2.
(b) If the first two throws resulted in red, what is the probability of red at the third throw?
(c) If red turns up at the first n throws, what is the probability that die A is being used?