## Review Exercises for Finite Mathematics Finite Mathematics & Applied Calculus Topic: Probability

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Enter your answer for each question in the left box, and press "Check." If you use decimals, they must be accurate to 4 decimal places. You can also use fraction notation, eg. 33/56.

1. In Advanced Carmine Canasta, in each player rools two dice (one red, one green), flips a coin, and selects three gummy bears from a package containing two rasberry ones, two lime ones and two orange ones.

 (a) How many outcomes are possible? (b) To have a winning conbination, both dice must show the same number, the coin toss must result in heads, and two of the gummy bears must be lime. What is the probability of winning Advanced Carmine Canasta?

2. The following table shows sales of luxury imported vehicles in the US in 1996-1998.

 Acura Infiniti Lexus Mercedes Total 1996 60,000 55,000 80,000 90,000 285,000 1997 70,000 65,000 95,000 120,000 350,000 1998 65,000 55,000 140,000 155,000 415,000 Total 195,000 175,000 315,000 365,000 1,050,000

Figures are rounded, and 1998 figures are annualized, based on sales through September. Source: Ward's Automobiles/The New York Times, October 21, 1998, p. C1.

Calculate the following (experimental) probabilities:

 (a) That a luxury imported vehicle purchased during 1996-1998 was an Acura (b) That a luxury imported vehicle purchased during during 1996-1998 was an Acura, given that it was an import from Japan (c) That a luxury vehicle was purchased after 1996, given that it was an import from Japan

3. A consumer survey reveals that the probability of a computer owner shopping on the Internet was 0.17, while the probability of a computer owner downloading software was 0.33. Further, the probability a computer owner doing both was 0.14. Find the probability of the following events:

 (a) that a computer owner does not shop on the Internet (b) that a computer owner will either shop on the Internet or download software (c) that a computer owner will neither shop on the Internet nor download software

4. The moon is in the Seventh House at the same time that Jupiter aligns with Mars 5% of the time, whereas Jupiter aligns with Mars 25% of the time. It is also found that, 75% of the time, either Jupiter aligns with Mars or the Moon is in the Seventh House, or both. What percentage of the time is the Moon in the Seventh House?

% of the time

5. Based on past statistics, 60% of finite math students who use this web site get high grades, whereas 15% of all finite math students use this web site and get a high grade. What is the probability that a finite math student uses this web site?

6. Two dice (one red, one green) are cast, and the uppermost numbers are observed. Test for independence (make a choice and press "Check"):

 (a) The red die is even; the green die is odd dependent independent (b) The red die is a 1; the sum is even dependent independent (c) Neither die is 1; the sum is even dependent independent (d) Both dice are 1; the sum is even dependent independent (e) Neither die is 1; both show the same number dependent independent

7. The moon is in the Seventh House once every 30 days on average, and when it is, Jupiter has a 10% chance of aligning with Mars; when it isn't, Jupiter has only a 5% chance of aligning with Mars.

 (a) What is the probability that the moon is in the Seventh House when Jupiter aligns with Mars? (b) What is the probability that the moon is in the Seventh House and Jupiter aligns with Mars?

M1. Give the transition matrix for each of the following.

P=

P=

M2. In each of the following, you are given a transition matrix P and an initial distribution vector v. Compute the quantities asked for.

(a) P=
 0.2 0.8 0.8 0.2
,   v = [100   0]

Probability of going from state 2 to state 1 in two steps =

Distribution vector after two steps is [   ]

(b) P=
 1/2 1/4 1/4 0 1 0 1/2 1/2 0
,   v = [0   160   0]

The two-step transition matrix is
The four-step transition matrix is
Probability of going from state 3 to state 1 in four steps =

Distribution vector after four steps is [     ]

M3. Reading Habits Jason Sauter is an avid reader of science fantasy, Proust novels and calculus texts. He visits the bookstore every week and buys one book. (He reads a book every week.) If he is presently reading Proust, there is a 50% chance that he will switch to calculus the next week, and he never reads two Proust volumes in a row. After a week of science fantasy, there is a 75% chance that he will switch to Proust the next week, and a 25% percent chance that he will switch to calculus. He always reads Proust after reading calculus. Assuming this trend continues indefinitely, * what fraction of his library will consist of science fiction 17,000 weeks from now?
science fiction

* Also assume that there is an inexhaustible supply of science fiction, Proust, and calculus books.

M4. Jones Beach Jones Beach has a jetty on its westernmost edge, a swimming area is on its easternmost edge, and a volleyball play area in between. One sunny day, you notice that one in six of the people in the volleyball area stroll to the jetty every hour, while two in six stroll over to the swim area. Due to the bad jellyfish infestation, half the beach goers at the swim area meander over to the volleyball area every hour, while the rest of them stay put. Nobody leaves the jetty.

(a) At noon, the entire beach population of 1,200 was in the swimming area. How crowded was the jetty by 2 pm.?
people

(b) What fraction of the people in the volleyball play area at noon will be at the jetty by 2 pm.?
at the jetty

M5. Malls Following is a plan of the "Petit Mall." As an potential store owner, you have been offered a lease at various store locations in the mall (the colored areas show the store locations), and are deciding which location to choose. Part of your decision will be based on a knowledge of shopper traffic in different parts of the mall (areas A through F).

During the first part of the morning hours, very few shoppers leave the mall through tghe exit door and the movie theater is closed. Further, shoppers tend to move randomly from one area to an adjacent area. For instance, a shopper in area B has a 1/3 chance of moving to A, C, or D, which a shopper in C is certain to move to B.

(a) If a shopper begins in location A when the mall opens, what is the probability of that shopper being found in location C after two changes of location (two time steps)?

(b) Calculate P2, P3, P4, and P5 by hand. On average, what percentage of shoppers can be found where they started after 5 changes of location?

M6. Sports Researchers have modeled the results of cricket games between England and Australia by the following Markov system, in which a time step corresponds to one game. (The states represent wins, losses and draws for England.) ††

 To Win Loss Draw From Win 0.4 0.4 0.2 Loss 0.3 0.4 0.3 Draw 0.3 0.3 0.4

In the long term, England can expect to win of the time, to lose of the time, and to draw of the time.

†† Figures are rounded to one decimal place. Source: Derek Colwell, Brian Jones and Jack Gillet, "A Markov Chain in Cricket," the Mathematical Gazette, June, 1991.

M7. Heating (Use of technology recommended. Try the the on-line matrix algebra tool for a direct computaiton, or the Pivot and Gauss-Jordan Tool to solve the associated system of equations.)
According to research by Gulf Oil Corporation, switching between oil, gas and electric heat in US homes followed the following pattern over a one-year period. Ý

 To Oil Gas Electric From Oil 0.825 0.175 0 Gas 0.060 0.919 0.021 Electric 0.049 0 0.951

What are the long term implications of these trends? (Round all percentages to two decimal places.)

% oil   % gas   % electric

Ý Source: Ali Ezzarti, "Forecasting Market Shares of Alternative Home Heating Units," Management Science, Vol. 21, no. 4, Dec. 1974. Copyright (c) 1974 by The Institute of Management Sciences.

M8. Neural Networks Consider the following model of a small neural network in which an impulse can travel from node to node with probabilities shown in the diagram.

(a) Write down the transition matrix P, and compute the steady state transition vector v.
(b) Is the given system regular?

M9. (Requires knowledge of absorbing systems) Petit Mall This continues the above question on the "Petit Mall."

During the evening hours, a shopper in Area C has a 0.5 probability of leaving the mall through Area E, while a shopper in area D has a 0.5 probability of going to the movies. Since the mall will be closed by the time the movie ends, regard areas E and F as absorbing states.

(a) Calcuate the fundamental matrix Q .

(b) How many visits will a shopper entering the mall in Area A be expected to make to Area D?

(c) Which area or areas will be visited the most frequently by a shopper entering the mall at Area A?

(d) What fraction of shoppers in Area D will ultimately go to the movies?

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Last Updated: July, 2006

Copyright © 1999, 2003, 2006 Stefan Waner and Steven R. Costenoble