Statistics Study Guide - Probability

Statistics Independent Study Page 1 of 5 Study Sheet Practice With Laws of Probability

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Review of Probability Concepts The probability of an event is the number of ways that event can occur divided by the total number of outcomes. Probabilities based on historical relative frequencies may be used to estimate the chance of an event occurring in the near future or in another similar population. Since a probability is a proportion, it takes values from zero to one. Probabilities can be calculated for equally likely outcomes by dividing the number of possible equally likely successes by the number of possible equally likely outcomes. However, knowing if outcomes are equally likely requires knowledge about the factors influencing the outcomes. When we have a given probability, we can compute the expected value, or the expected relative frequency, for an event with a given number of trials or size of sample. The observed value will get closer to the expected value as the number of trials or the sample size gets larger. The law of large numbers says that error due to chance (the difference between the observed and expected values for a number of events) decreases as the sample size or number of trials increases. When solving a probability problem, the basic approach is to construct the sample space for the experiment under consideration. To answer your problem, all you have to do is count the number of successful outcomes as a fraction of the total outcomes in the sample space. Events are composed of simple events. Simple events are events that can't be broken down into smaller events. Two different events can have some outcomes in common, but mutually exclusive, or disjoint, events have no outcomes in common. The sample space is the collection of all possible outcomes. The complement of an event is the outcomes in the sample space that aren't part of the event. So, the sum of the probability of an event and the probability of its complement is always one. Two events are independent if the occurrence of one event doesn't affect the probability of the other event occurring. A Venn Diagram is a graphical representation of the sample space and events. Conditional probability refers to the probability of some event occurring, given knowledge of some other event that may influence its probability. We refer to the condition "the probability of an event A, given that event B has occurred" symbolically as P(A | B). General probability rules For any two events A and B P(A and B) = P(A) P(B | A) P(A or B) = P(A) + P(B) P(A and B)



Specific probability rules from the general Since P(B) = P(B | A) if A and B are independent, the rule P(A and B) = P(A) P(B | A) becomes P(A and B) = P(A)P(B). Since P(A and B) = 0 if A and B are mutually exclusive, the rule P(A or B) = P(A) + P(B) P(A and B) becomes P(A or B) = P(A) + P(B). Questions 1. An experiment involves tossing a single die. These are some events:

A. Observe a 2

B. Observe an even number

C. Observe a number greater than 2

D. Observe both A and B

E. Observe A or B or both

F. Observe both A and C

a. List the simple events in the sample space.

b. List the simple events in each of the events A through F.

c. What probabilities should you assign to the simple events?

d. Calculate the probabilities of the six events A through F by adding the appropriate simple-event probabilities.

2. A particular player hits 70% of her free throws. When she tosses a pair of free

throws, the four possible events and three of their associated probabilities are as given in the table:

Simple Event

Outcome of First Free Throw

Outcome of Second Free Throw

Probability

1 Hit Hit .49 2 Hit Miss ? 3 Miss Hit .21 4 Miss Miss .09

A. Find the probability that the player will hit on the first throw and miss

on the second. B. Find the probability that the player will hit on at least one of the two

free throws.



3. Three people are randomly selected from a voter registration and driving records to report for jury duty. The gender of each person is noted by the county clerk.

A. List the simple events in S.

B. If each person is just as likely to be a man as a woman, what probability do you assign to each simple event?

C. What is the probability that only one of the three is a man?

D. What is the probability that all three are women?

4. Four equally qualified runners, John, Bill, Ed, and Dave, run a 100-meter sprint, and

the order of finish is recorded.

A. How many simple events are in the sample space?

B. If the runners are equally qualified, what probability should you assign to each simple event?

C. What is the probability that Dave wins the race?

D. What is the probability that Dave wins and John places second?

E. What is the probability that Ed finishes last?

5. In a genetics experiment, the researcher mated two Drosophila fruit flies and

observed the traits of 300 offspring. The results are shown in the table.

Wing Size Eye Color Normal Miniature

Normal 140 6 Vermillion 3 151

One of these offspring is randomly selected and observed for the two genetic traits.

A. What is the probability that the fly has normal eye color and normal wing size?

B. What is the probability that the fly has vermillion eyes?

C. What is the probability that the fly has either vermillion eyes or miniature wings, or both?



6. An experiment consists of tossing a single die and observing the number of dots that show on the upper face. Events A, B, and C are defined as follows:

A. Observe a number less than 4 B. Observe a number less than or equal to 2 C. Observe a number greater than 3

Find the probabilities associated with these compound events using either the simple event approach or the rules and definitions from this section.

a. S

b. A|B

c. B

d. A B C

e. A B

f. A C

g. B C

h. A C

i. B C

7. Refer to Question 6.

A. Are events A and B independent? Mutually exclusive?

B. Are events A and C independent? Mutually exclusive?

8. Suppose that P(A) = .4 and P(B) = .2. If events A and B are independent,

find these probabilities:

A. P(A B)

B. P(A B)

9. A certain manufactured item is visually inspected by two different

inspectors. When a defective item comes through the line, the probability that it gets by the first inspector is .1. Of those that get past the first inspector, the second inspector will "miss" five out of ten. What fraction of the defectives gets by both inspectors?



10. A smoke-detector system uses two devices, A and B. If smoke is present, the probability that it will be detected by device A is .95; by device B, .98; and by both devices, .94.

A. If smoke is present, find the probability that the smoke will be detected by device A or device B, or both devices.

B. Find the probability that the smoke will not be detected.

Acknowledgements Question 1: This is question 4.1 (a, b, c, d) from page 126 of Introduction to Probability and Statistics, Tenth Edition, by W. Mendenhall, R. Beaver, and B. Beaver. Copyright 1999 by Brooks Cole, division of Thompson Learning Incorporated. Further reproduction is prohibited without permission of the publisher. Question 2: This is question 4.4 (a, b,) from page 127 of Introduction to Probability and Statistics, Tenth Edition, by W. Mendenhall, R. Beaver, and B. Beaver. Copyright 1999 by Brooks Cole, division of Thompson Learning Incorporated. Further reproduction is prohibited without permission of the publisher. Question 3: This is question 4.11 (b, c, d, e) from page 128 of Introduction to Probability and Statistics, Tenth Edition, by W. Mendenhall, R. Beaver, and B. Beaver. Copyright 1999 by Brooks Cole, division of Thompson Learning Incorporated. Further reproduction is prohibited without permission of the publisher. Question 4: This is question 4.14 (a, b, c, d, e) from page 129 of Introduction to Probability and Statistics, Tenth Edition, by W. Mendenhall, R. Beaver, and B. Beaver. Copyright 1999 by Brooks Cole, division of Thompson Learning Incorporated. Further reproduction is prohibited without permission of the publisher. Question 5: This is question 4.15 (a, b, c) from page 129 of Introduction to Probability and Statistics, Tenth Edition, by W. Mendenhall, R. Beaver, and B. Beaver. Copyright 1999 by Brooks Cole, division of Thompson Learning Incorporated. Further reproduction is prohibited without permission of the publisher. Question 6: This is question 4.45 (a, b, c, d, e, f, g, h, i) from page 148 of Introduction to Probability and Statistics, Tenth Edition, by W. Mendenhall, R. Beaver, and B. Beaver. Copyright 1999 by Brooks Cole, division of Thompson Learning Incorporated. Further reproduction is prohibited without permission of the publisher. Question 7: This is question 4.46 (a, b) from page 148 of Introduction to Probability and Statistics, Tenth Edition, by W. Mendenhall, R. Beaver, and B. Beaver. Copyright 1999 by Brooks Cole, division of Thompson Learning Incorporated. Further reproduction is prohibited without permission of the publisher. Question 8: This is question 4.47 (a, b) from page 148 of Introduction to Probability and Statistics, Tenth Edition, by W. Mendenhall, R. Beaver, and B. Beaver. Copyright 1999 by Brooks Cole, division of Thompson Learning Incorporated. Further reproduction is prohibited without permission of the publisher. Question 9: This is question 4.59 from page 151 of Introduction to Probability and Statistics, Tenth Edition, by W. Mendenhall, R. Beaver, and B. Beaver. Copyright 1999 by Brooks Cole, division of Thompson Learning Incorporated. Further reproduction is prohibited without permission of the publisher. Question 10: This is question 4.61 (a, b) from page 151 of Introduction to Probability and Statistics, Tenth Edition, by W. Mendenhall, R. Beaver, and B. Beaver. Copyright 1999 by Brooks Cole, division of Thompson Learning Incorporated. Further reproduction is prohibited without permission of the publisher.

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Statistics Study Guide - Probability