MATH231_EXAM2

Permutation

"Any arrangement of r objects selected from a single group of n possible objects." n P r = n! / (n − r)!" n is the total number of objects r is the number of objects selected

A golfer has 12 golf shirts in his closet. Suppose 9 of these shirts are white and the others blue. He gets dressed in the dark, so he just grabs a shirt and puts it on. He plays golf two days in a row and does not launder and return the used shirts to the closet. What is the likelihood both shirts selected are white?

(9/12)(8/11) = .55

"In a management trainee program at Claremont Enterprises, 80% of the trainees are female and 20% male. Ninety percent of the females attended college, and 78% of the males attended college. a. A management trainee is selected at random. What is the probability that the person selected is a female who did not attend college? b. Are gender and attending college independent? Why?

.80 * .10 = .08 not attend college No, if you know the gender it will change the probability if they attended college

If the set of events is collectively exhaustive and the events are mutually exclusive, the sum of the probabilities is

1

"The first card selected from a standard 52-card deck is a king. a. If it is returned to the deck, what is the probability that a king will be drawn on the second selection? b. If the king is not replaced, what is the probability that a king will be drawn on the second selection?" c. What is the probability that a king will be selected on the first draw from the deck and another king on the second draw (assuming that the first king was not replaced)?"

1/13 =.769 3/31 = .058 1/13 * 1/17 = 1/221 = .0045

Contingency Table

A table used to classify sample observations according to two or more identifiable categories or classes. summarizes two variables of interest and their relationship.

Collectively exhaustive

At least one of the events must occur when an experiment is conducted."

Special Rule of Multiplication

For two independent events A and B, the probability that A and B will both occur P(A and B) = P(A)P(B)

General Rule of Multiplication

P(A and B) = P(A)P(B | A )

The events A and B are mutually exclusive. Suppose P(A) = .30 and P(B) = .20. What is the probability of either A or B occurring? What is the probability that neither A nor B will happen?

P(A or B) = P(A) + P(B) = .30 + .20 = .50 P(neither) = 1 − .50 = .50.

General Rule of Addition

P(A or B) = P(A) + P(B) − P(A and B) Ex: What is the probability that a card chosen at random from a standard deck of cards will be either a king or a heart? "add the probability of a king (there are 4 in a deck of 52 cards) to the probability of a heart (there are 13 in a deck of 52 cards) and report that 17 out of 52 cards meet the requirement, we have counted the king of hearts twice. We need to subtract 1 card from the 17 so the king of hearts is counted only once. Thus, there are 16 cards that are either hearts or kings. So the probability is 16/52 = .3077." P(A or B) = P(A) + P(B) − P(A and B) = 4/52 + 13/52 − 1/52 = 16/52 = .3077

The probabilities of the events A and B are .20 and .30, respectively. The probability that both A and B occur is .15. What is the probability of either A or B occurring?

P(A or B) = P(A) + P(B) − P(A and B) = .20 + .30 − .15 = .35"

"On February 1, 2003, the Space Shuttle Columbia exploded. This was the second disaster in 113 space missions for NASA. On the basis of this information, what is the probability that a future mission is successfully completed?"

P(A)=111/113 = .98

"Brooks Insurance Inc. wishes to offer life insurance to men age 60 via the Internet. Mortality tables indicate the likelihood of a 60-year-old man surviving another year is .98. If the policy is offered to five men age 60: a. What is the probability all five men survive the year? b. What is the probability at least one does not survive?"

P(AandB) = P(A)P(B) =(.98)(.98)(.98)(.98) =.0961 =1-P(~A) =1-.9039 =.0961

"There are three clues labeled "daily double" on the game show Jeopardy. If three equally matched contenders play, what is the probability that: a. A single contestant finds all three "daily doubles"? b. The returning champion gets all three of the "daily doubles"? c. Each of the players selects precisely one of the "daily doubles"?"

P(B and ~A) = P(~A)P(B|~A) 29/30 * 1/3 = 29/30 first 1/3 * 1/3 = 1/9 second 29/90 * 1/9=29/810 - .0388 p=(1/30+29/30)*1/9 =32/810 =.0395 P(1/30+29/90)(1/3*1/3 + 1/3*1/3) =32/90*2/9 = 32/405 first p=29/90(1/3*1/3 + 1/3*1/3) =29/405 second p=32/405+29/405 =61/405 =.1506 total

"A survey by the American Automobile Association (AAA) revealed 60% of its members made airline reservations last year. Two members are selected at random. What is the probability both made airline reservations last year?"

P(R 1 and R 2 ) = P(R 1 )P(R 2 ) = (.60)(.60) = .36

Suppose the probability you will get an A in this class is .25 and the probability you will get a B is .50. What is the probability your grade will be above a C?

P(above C) = .25 + .50 = .75

Complement rule

Probability of an event occurring by subtracting the probability of the event not occurring from 1 P(A) = 1 − P(∼A) Ex: The probability a bag of mixed vegetables is underweight is .025 and the probability of an overweight bag is .075. Use the complement rule to show the probability of a satisfactory bag is .900

Subjective probability

The likelihood of a particular event happening based on whatever information is available -> everything else is Objective probability

Independent

The occurrence of one event has no effect on the probability of the occurrence of another event.

Mutually exclusive

The occurrence of one event means that none of the other events can occur at the same time. Ex: "female. An employee selected at random is either male or female but cannot be both."

Conditional probability

The probability of a particular event occurring, given that another event has occurred.

Empirical probability

The probability of an event happening is the fraction of the time similar events happened in the past. (# of times event occurs / total # of observations)

Special Rule of addition

events must be mutually exclusive P(A or B) = P(A) + P(B) Ex: "What is the probability that a particular package will be either underweight or overweight?"

Classical probability

is based on the assumption that the outcomes of an experiment are equally likely. Number of favorable outcomes divided by the number of possible outcomes Ex: "There are 3 "favorable" outcomes in the collection of 6 equally likely possible outcomes. 3/6 = 5

r

measure of strength relationship between 2 variables

"Referring to the group of three electronic parts that are to be assembled in any order, in how many different ways can they be assembled?"

r = 3. n P r = n!/(n-r) = 3!/(3-3) = 3!/0 =3!/1 = 6

Joint Probablity

two events occur at the same time uses the general rule of addition Ex: The prob- ability (.30) that a tourist visits both attractions is an example of a joint probability.