Posted: November 11th, 2015

Maths

Maths

1. An orthodontist has three financing packages, and each has a different service charge.  He estimates that 30% of patients use the first plan, which has a $10 finance charge: 50% use the second plan, which as a $20 finance charge; and 20% use the third plan, which has a $30 finance charge.
a) Find and expected value of the service charge. $19.
b) Find the standard deviation of the service charge. $7.
2. A company selling glass ornaments by mail-order expects, from previous history, that 6% of the ornaments it ships will break in shipping.  You purchase two ornaments as gifts and have them shipped separately to tow different addresses.  What is the probability that both arrive safely?  What did you assume?
3. Help Desk: The computer help desk is staffed by students during the 7:00PM to 11:00Pm shift.  Let Y denote the random variable that represents the number of students seeking help during the 15-minute time slot 10:00 to 10:15 P.M.
a) What are the possible values of Y?
b) Is the random variable discrete or continuous?
4. Fishing tournament: A sporting goods manufacturer was asked to sponsor a local boy in two fishing tournaments.  They claim the probability that he will win the first tournament is 0.4.  If he wins the first tournament they estimate the probability that he will also win the second is 0.2.  The guess that if he loses the first tournament, the probability that he will win the second is 0.3.
a) According to their estimates, are the two tournaments independent?  Explain your answer. They are not independent.  Probability of winning the second tournament changes if he wins the first tournament.
b) What’s the probability that he loses both tournaments? 0.42
c) What’s the probability he wins both tournaments? 0.08
d) Let random variable X be the number of tournaments he wins.  Find the probability              model for X.
X    0    1    2
P(X = x)    0.42    0.50    0.08

e) What are the expected value and standard deviation of X ?  E(X) = 0.66 tournaments s = 0.62 tournaments
5. Contracts:  Your company bids for two contracts.  You believe the probability that you get contract #1 is 0.8.  If you get contract #1, the probability that you also get contract #2 will be 0.2, and if you do not get contract #1, the probability that you get contract #2 will be 0.3.
a) Are the outcomes of the two contract bids independent?  Explain.
b) Find the probability you get both contracts.
c) Find the probability you get neither contracts.
d) Let X be the number of contracts you get.  Find the probability model for X.

6. Grocery supplier:  A grocery supplier believes that the mean number of broken eggs per dozen is 0.6, with a standard deviation of 0.5.  you buy 3 dozen eggs without checking them.
a) How many broken eggs do you expect to get?
b) What’s the standard deviation?
c) Is it necessary to assume the cartons of eggs are independent?  Why?
7. Casino:  At a casino, people play the slot machines in hopes of hitting the jackpot, but most of the time, they lose their money.  A certain machine pays out an average of $0.92 (for every dollar played), with a standard deviation of $120.
a) Why is the standard deviation so large?
b) If a gambler plays 5 times, what are the mean and standard deviations of the casino’s profit?
c) If gamblers play this machine 1000 times in a day, what are the mean and standard deviation of the casino’s profit?
8. Closing costs: A salesman normally makes a sale (closes) on 80% of his presentations.  Assuming the presentations are independent, find the probability of each of the following.
a) He fails to close for the first time on his fifth attempt.  0.0819
b) He closes his first presentation on his fourth attempt.  0.00064
c) The first presentation he closes will be on his second attempt.  0.16
d) The first presentation he closes will be on one of his first three attempts.  0.992
9. Professional tennis: Serena Williams made a successful first serve 67% of the time in a Wimbledon finals match against her sister Venus.  If she continues to serve at the same rate the next time they play and serves 6 times in the first game, determine the following probabilities.  (Assume that each serve is independent of the others)
a) All 6 first serves will be in.  0.090
b) Exactly 4 first serves will be in.  0.329
c) At least 4 first serves will be in.  0.687
10. Web visitors: A website manager has noticed that during the evening hours, about 3 people per minute check out from their shopping cart and make an online purchase.  She believes that each purchase is independent of the others and wants to model the number of purchases per minute.
a) What model might you suggest to model the number of purchases per minute?  The Poisson model.
b) What is the probability that in any one minute at least on purchase is made? 0.8502
c)What is the probability that no one makes a purchase in the next 2 minutes?   0.0025

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