1. Bill is playing mini-golf. We meet up with him at the 13th hole, where he makes his second hole-in-one. Assume that the probability that Bill makes a hole-in-one is 1/10 for every hole.

   (a) What is the chance that Bill’s first hole-in-one was made in the 4th hole?
   (b) Let X be the number of the hole Bill made the first hole-in-one. Find the PMF of X.

Tom works in a shop. On average, he sees that 3 purchasers come to the shop every hour. Calculate the following:

a) What is the probability that at least 2 and as many as 4 purchasers come to the shop in a given hour?

b) The mean rate of purchaser decreases to 5 purchasers every 2 hours. During one hour, what is the most likely number of purchasers that will come to the shop (hint: the answer must be an integer)?

Question 2. Draw three cards without replacement from a deck of cards and let X be the number of spades drawn. Sketch the pmf of X and compute E(X).

Question 3. A fair coin is flipped n times. What is the probability of getting a total of k heads if

a) The first flip shows heads

b) The first flip shows tails

c) At least one flip shows heads

An urn has 8 red and 12 blue balls. Suppose that balls are chosen at random and removed from the
urn, with the process stopping when all the red balls have been removed. Let X be the number of balls
that have been removed when the process stops.
a. Find P(X = 14)
b. Find the probability that a specifed blue ball remains in the urn.
c. Find E[X]

16) For planning purposes, insurance companies must try to predict the number of hurricanes that will hit the eastern coast of the United States in any given year. If this area is hit with an average of 6 hurricanes per year, find the probability that in a given year…

a) …fewer than 3 hurricanes will hit this area.

b) …between 6 and 8 hurricanes (inclusive) will hit this area.

c) What is the standard deviation of this distribution?

Three students answer a true-false question on a test randomly. A random variable x represents the number of answers of true among the three students, taking on the value 0, 1, 2, or 3.

(a) Find the frequency distribution for x.
(b) Find the probability distribution for x.

For the same random variable x as the previous problem, find the following:

(a) The expected value (b) The variance (c) The standard deviation

A recent survey found that 63% of all adults over 50 wear glasses for driving. Ia random sample of 100 adults over 50 was selected., a). what is the expected value and standard deviation of the number of adults over 50 who wear glasses? b). Find the probability that more than 60 of the adults over 50 wear glasses. Hint: using normal distribution to approximate binomial distribution.

sixty four percent of U.S. adults oppose hydraulic fracturing​ (fracking) as a means of increasing the production of natural gas and oil in the United States. You randomly select six U.S. adults. Find the probability that the number of U.S. adults who oppose fracking as a means of increasing the production of natural gas and oil in the United States is​ (a) exactly three​, ​(b) less than four​, and​ (c) at least three.

9. Suppose that 32% of the apples grown on a particular orchard have blemishes that make it difficult for them to be sold at a market. If a market purchases a crate that contains 100 apples selected randomly from the orchard, answer each of the following.

   1. Find the mean and standard deviation for the number of blemished apples in the crate.

   2. Compute the probability of there being exactly 32 blemished apples in the crate.

   3. Would it be considered unusual for 40 or more of the apples to be blemished?

Urn A contains 5 green and 4 red balls, and Urn B contains 3 green and 6 red balls. One ball is drawn from Urn A and transferred to Urn B. Then one ball is drawn from Urn B and transferred to Urn A. Let X = the number of green balls in Urn A after this process. List the possible values for X and then find the entire probability distribution for X.