Theory Chapter 5 Business Statistics

An automobile dealer wants to advertise that for $29,999 you can buy a convertible, a two-door sedan, or a four- door model with your choice of either wire wheel covers or solid wheel covers. How many different arrangements of models and wheel covers can the dealer offer?

(3)(2)=6

26)A sample of executives were surveyed about their loyalty to their company. One of the questions was, "If you were given an offer by another company equal to or slightly better than your present position, would you remain with the company or take the other position?" The responses of the 200 executives in the survey were cross-classified with their length of service with the company.(word)

(word)

27) (word)

(word)

Conditional probability

- A conditional probability is the probability of a particular event occurring, given that another event has occurred. - The probability of the event A given that the event B has occurred is written P(A|B).

What is a factorial?

- A factorial is a function that multiplies a number by every number below it. For example 5!= 5*4*3*2*1=120. - The function is used, among other things, to find the number of way "n" objects can be arranged. - 4factorialis4!=4x3x2x1=24 --> There are 24 different ways to arrange the numbers 1 through 4. {1,2,3,4}, {2,1,3,4}, {2,3,1,4}, {2,3,4,1}, {1,3,2,4}, etc.

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 do laundry. What is the likelihood both shirts selected are white?

- The probability that the first shirt selected is white is P(W1) = 9/12. - The probability of selecting a second white shirt (W2 ) is dependent on the first selection. So, the conditional probability is the probability the second shirt selected is white, given that the first shirt selected is also white: P(W2 | W1) = 8/11. - Apply the General Multiplication Rule: P(A and B) = P(A) P(B|A) - The joint probability of selecting 2 white shirts is: P(W1 and W2) = P(W1)P(W2 |W1) = (9/12)(8/11) = 0.55

Special rule of multiplication

- The special rule of multiplication calculates the joint probability of two events A and B that are independent. - Two events A and B are independent if the occurrence of one has no effect on the probability of the occurrence of the other. - This rule is written: P(A and B) = P(A)P(B)

24)word

1)Based on the empirical information, P(zero movies) = 60/150 = 0.4 2)Applying the General Rule of Addition: P( zero movies or male) = P( zero movies) + P(male) - P(zero movies and male) P(zero movies or male) = 60/150 + 70/150 - 20/150 = 0.733 3)Applying the concept of conditional probability: P( zero movies | male) = 20/70 = 0.286 4)Applying the General Rule of Multiplication P( male and zero movies) = P(male)P(zero movies|male) = (70/150)(20/70) = 0.133

There are three ways to assign probability?

1. CLASSICAL PROBABILITY Based on the assumption that the outcomes of an experiment are equally likely. 2. EMPIRICAL PROBABILITY The probability of an event happening is the fraction of the time similar events happened in the past. 3. SUBJECTIVE PROBABILITY The likelihood (probability) of a particular event happening that is assigned by an individual based on whatever information is available.

32) Combination

A combination is the number of ways to choose r objects from a group of n objects without regard to order.(word)

What is a contingency table?

A contingency table is used to classify sample observations according to two or more identifiable characteristics measured.

31)Permutation

A permutation is any arrangement of r objects selected from n possible objects. The order of arrangement is important in permutations.(word)

Tree diagram?

A tree diagram is: 1. Useful for portraying conditional and joint probabilities. 2. Particularly useful for analyzing business decisions involving several stages. 3. A graph that is helpful in organizing calculations that involve several stages. Each segment in the tree is one stage of the problem. The branches of a tree diagram are weighted by probabilities.

What is a probability?

A value between zero and one, inclusive, describing the relative possibility (chance or likelihood) an event will occur.

Event

An event is the collection of one or more outcomes of an experiment

Experiment

An experiment is a process that leads to the occurrence of one and only one of several possible results

Outcome

An outcome is the particular result of an experiment

Example classical probability

Consider an experiment of rolling a six-sided die. What is the probability of the event: "an even number of spots appear face up"? There are three "favorable" outcomes (a two, a four, and a six) in the collection of six equally likely possible outcomes.

Rules of addition for computing probabilities

If two events A and B are mutually exclusive, the probability of one or the other event occurring equals the sum of their probabilities. P(A or B) = P(A) + P(B)

Example of subjective probability

Illustrations of subjective probability are: 1. Estimating the likelihood the New England Patriots will play in the Super Bowl next year. 2. Estimating the likelihood a person will be married before the age of 30. 3. Estimating the likelihood the U.S. budget deficit will be reduced by half in the next 10 years.

The General Rule of Addition - If A and B are two events that are not mutually exclusive, then P(A or B) is given by the following formula:

P(A or B) = P(A) + P(B) - P(A and B) P( A and B) is called a joint probability.

17) What is the probability that a card chosen at random from a standard deck of cards will be either a king or a heart? (word)

P(AorB)= P(A)+P(B)-P(AandB) = 4/52 + 13/52 - 1/52 = 16/52, or .3077

14)A machine fills plastic bags with a mixture of beans, broccoli, and other vegetables. Most of the bags contain the correct weight, but because of the variation in the size of the beans and other vegetables, a package might be underweight or overweight. A check of 4,000 packages filled in the past month revealed: (word) What is the probability that a particular package will be either underweight or overweight?

P(AorC)=P(A)+P(C) =.025+.075=.10 Note that P(A or C) = P(~B), so P(~B) = 1 - P(B) = 1 - .900 = .10

An experiment has two mutually exclusive outcomes. Based on the rules of probability, the sum of the probabilities must be one. If the probability of the first outcomes is .61, then logically, and by the complement rule, the probability of the outcome is ?

P(B) = 1 - P(~B) = 1 - .61 = .39

The Venn Diagram shows the results of a survey of 200 tourists who visited Florida during the year. The results revealed that 120 went to Disney World, 100 went to Busch Gardens, and 60 visited both. What is the probability a selected person visited either Disney World or Busch Gardens?

P(Disney or Busch) = P(Disney) + P(Busch) - P(both Disney and Busch) = 120/200 + 100/200 - 60/200 = .60 + .50 - .80

Example empirical probability On February 1, 2003, the Space Shuttle Columbia exploded. This was the second disaster in 123 space missions for NASA. On the basis of this information, what is the probability that a future mission is successfully completed?

Probability of a successful flight= Number of successful flights/Total number of flights=121/123=0.98

A survey by the American Automobile Association (AAA) revealed 60 percent of its members made airline reservations last year. Two members are selected at random. Since the number of AAA members is very large, we can assume that R1 and R2 are independent. What is the probability both made airline reservations last year?

Solution: The probability the first member made an airline reservation last year is .60, written as P(R1) = .60 The probability that the second member selected made a reservation is also .60, so P(R2) = .60. P(R1 and R2) = P(R1)P(R2) = (.60)(.60) = .36

Complement rule

The complement rule is used to determine the probability of an event occurring by subtracting the probability of not occurring from 1. P(A)+P(-A)=1 or P(A)=1-P(-A)

20) General Rule of Multiplication

The general rule of multiplication is used to find the joint probability that two events will occur when they are not independent. (word) It states that for two events, A and B, the joint probability that both events will happen is found by multiplying the probability that event A will happen by the conditional probability of event B occurring given that A has occurred.

subjective probability?

The likelihood (probability) of a particular event happening that is assigned by an individual based on whatever information is available. If there is little or no data or information to calculate a probability, it may be arrived at subjectively.

Counting Rules - Multiplication

The multiplication formula indicates that if there are m ways of doing one thing and n ways of doing another thing, there are m x n ways of doing both. Example: Dr. Delong has 10 shirts and 8 ties. How many shirt and tie outfits does he have? (10)(8) = 80

Empirical probability

The probability of an event happening is the fraction of the time similar events happened in the past. Empirical approach to probability is based on what is called the Law of Large Numbers. Law of large numbers: Over a large number of trials, the empirical probability of an event will approach its true probability. The key to establishing probabilities empirically: a larger number of observations provides a more accurate estimate of the probability.

33)There are 12 players on the Carolina Forest High School basketball team. Coach Thompson must pick five players among the twelve on the team to comprise the starting lineup. How many different groups are possible?

word

34)There are 12 players on the Carolina Forest High School basketball team. Coach Thompson must pick five players among the twelve on the team to comprise the starting lineup. How many different groups are possible?

word