Probability Notes

 

3.1

 

Probability—the likelihood that an event will occur.

 

Consider the following example: During class, Dr. Rivera randomly selected a student and asked whether the student’s major was early childhood or middle grades.

 

Ask yourself the following questions:
1. How many answers did the student give? (one)

2. What were the possible answers? (early childhood or middle grades)

 

When a process results in just one answer, we call it an experiment.

 

Experiment—a process that when performed results in one and only one of many observations.

 

Outcomes—the observations of the experiment.

 

Sample Space—the collection of all outcomes for an experiment; denoted by S.

 

From the example,

  • Experiment: selecting a student and asking major
  • Outcomes: early childhood, middle grades
  • Sample space: S = {early childhood, middle grades}

 

Venn diagrams and tree diagrams are often used to represent the sample space of an experiment.  The Venn diagram and tree diagram for this example are below.

                                                S

                                                           

 

 

Suppose now that the experiment is changed to asking two students for their majors.  What are the possible outcomes of this experiment?  To make this easier, let’s abbreviate early childhood as EC and middle grades as MG.  The outcomes are EC EC, EC MG, MG EC, and MG MG.

 

Event—a collection of one or more of the outcomes of the experiment.

 

Example: For this new experiment an event might be that both students selected were middle grades majors.  Another event might be that at least one of the students was an early childhood major.

 

Simple event—an event that includes one and only one of the final outcomes for an experiment.

 

Example: The event that both students selected are middle grades majors is a simple event.  Of all of the possible outcomes (i.e. EC EC, EC MG, MG EC, MG MG) the only outcome described by this simple event is MG MG.

 

Compound event—a collection of more than one outcome for an experiment.

 

Example: The event that at least one of the students is an early childhood major is a compound event.  Of all of the possible outcomes (i.e. EC EC, EC MG, MG EC, MG MG), the outcomes EC EC, EC MG, MG EC are described by this compound event.

 

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3.2

 

A Random Event is one in which we are not sure of exactly what will happen even though we know the possible outcomes and we can predict a long term pattern.

 

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3.3/3.4

 

Probability—a numerical measure of the likelihood that a specific event will occur; if the event is A, then the probability that A will occur is denoted P(A).

 

Example: Flip a coin.  What is the probability of heads?  This is denoted P(heads).

 

Properties of Probability

 

  1. The probability of an event always lies in the range of zero to one.

 

            Impossible event—an event that absolutely cannot occur; probability is zero.

            Example: Suppose you roll a normal die.  What is the probability that you will get           a seven?  P(7) = 0.

            Sure event—an event that is certain to occur; probability is one.

            Example: Suppose you roll a normal die.  What is the probability that you will get           a number less than 7?  P(a number less than 7) = 1.

 

  1. The sum of the probabilities of all simple events (or final outcomes) for an experiment is always one.

 

            Example: Suppose you flip two coins.  What are the outcomes?  HH, HT, TH, TT

            This rule says that the probabilities of each of these outcomes should sum to one.                       That is,

                        P(HH) + P(HT) + P(TH) + P(TT) = 1

 

 

Two Types of Probability

 

  1. Experimental Probability—probability estimates arrived at using data gathered through experiments.
  2. Theoretical Probability—the specific value that is approached by experimental probability as more experiments are run.

 

Three Approaches to Calculating Probability

 

  1. Classical Probability
    • In this approach, outcomes are assumed to be equally likely.
    • Equally likely outcomes—two or more outcomes (or events) that have the same probability of occurrence.
    • Example: When you flip a fair coin, the probability of getting heads is equal to the probability of getting tails.  Heads and Tails are equally likely outcomes.
    • If the outcomes are equally likely, then we use the classical probability rule to calculate the probability.
    • Classical Probability Rule—P(A) = (# of successes)/(Total # of outcomes)
    • Example: Suppose you roll a die.

§         P(5) = 1/6

§         P(odd) = 3/6 = 1/2

§         P(not 4) = 5/6

  1. Relative Frequency
    • If the outcomes for an experiment are not equally likely, we cannot use the Classical Probability Rule.
    • Instead, we perform the experiment again and again to generate data.  We use the relative frequencies from this data to approximate the probability.
    • Example: What is the probability that the next car that comes out of an auto factory is a “lemon?”  In this experiment, the outcomes are “lemon” and “not lemon.”  Clearly, these two outcomes are not equally likely.  If they were, then half of the cars produced would be a lemon.  So, to find the probability of a lemon we must conduct an experiment repeatedly and gather some data.  Suppose we go to the auto plant and check the next 500 cars to see if they are lemons.  We find that 10 out of the 500 are lemons.  Using this information, we can approximate the probability that a car is a lemon using the relative frequency.

                                          P(lemon) = 10/500 = .02

    • Relative Frequency as an Approximation of Probability—n = # of times you repeat the experiment; f = # of times an event A is observed; P(A) = f/n.
    • From the example, the probability that we computed is not exact.  It is only an approximation.  How could we get a better approximation?  Check a larger number or cars. 
    • Law of Large Numbers—if an experiment is repeated again and again, the probability of an event obtained from the relative frequency approaches the actual or theoretical probability.
  1. Subjective Probability
    • Consider this question: What is the probability that Auburn will beat Georgia this year?  The question does not represent equally likely outcomes.  Nor does it represent an experiment that can be conducted over and over again.  When this happens, we have to use subjective probability.
    • Subjective probability—the probability assigned to an event based on subjective judgment, experience, information, and belief.
    • For the above question, Dr. Rivera may say the probability is .95 or 95% whereas someone else (a Georgia fan perhaps) may say the probability is 0, i.e. that it is not going to happen.

 

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Marginal and Conditional Probabilities

 

Suppose the faculty at a local school were polled as to their agreement/disagreement with the following statement: Coaches should be paid more than regular classroom teachers. 

 

The following two-way table contains the results.

                                   

 

Agree

disagree

MALE

20

10

FEMALE

15

35

 

From such a table, we can compute two types of probability—marginal and conditional.  First, you should add a row and column to the table for totals.

 

                                               

 

Agree

disagree

total

MALE

20

10

30

FEMALE

15

35

50

TOTAL

35

45

80

 

 

Marginal Probability—the probability of a single event without consideration of any other event; also called simple probability.

 

Example: P(male) = (# of males)/(total # of teachers) = 30/80

Example: P(agree) = (# of teachers who agree)/(total # of teachers) = 35/80

 

Conditional Probability—the probability that an event will occur given that another event has already occurred.

 

Example: Suppose that one teacher is selected at random.  It is known that the teacher is a male.  What is the probability that the teacher agrees?

.

 is read “the probability the teacher agrees given that the teacher is a male.”

 

Example: .

 

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3.5

 

Geometric Probability

 

  • In some cases it is not possible to compute the theoretical probability of an event by counting successes over total.
  • Geometric probability is another approach to computing probability.

 

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Fair Games, Odds, & Counting

 

Odds—the ratio of the probability that an event will occur to the probability that the event will not occur.

 

Example: Flip a coin twice.  What are the odds of getting at least one head?

Solution: We need the ratio P(at least one head)/P(no heads).  That is (3/4)(1/4) which is 3:1.

 

Rule: Given the odds of an event occurring is m:n, the probability of that even occurring is m/(m + n).

 

Example: The odds of winning a game are 1:45.  What is the probability of winning?

Solution: P(winning) = 1/(45 + 1) = 1/46.

 

The Counting Rule—if an experiment consists of three steps and if the first step can result in m outcomes, the second step in n outcomes, and the third step in k outcomes, then the total number of outcomes for the experiment is m*n*k.  This rule can be extended to experiments with more or less steps.

 

Example: If a coin is flipped 3 times, how many outcomes are possible?

Solution: Each time the coin is flipped, there are 2 outcomes (heads or tails).  2*2*2 = 8 total outcomes.

 

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Mutually Exclusive Events & Independent/Dependent Events

 

Mutually Exclusive Events—events that cannot occur together.

 

Example: A card is chosen at random from a standard deck of 52 playing cards.  Consider the event “the card is a 5” and the event “the card is a king.”  These two events are mutually exclusive because there is no way for both events to occur at the same time.

 

Non-example: A card is chosen at random from a standard deck of 52 playing cards.  Consider the event “the card is a heart” and the event “the card is a king.”  These two events are not mutually exclusive because it is possible for these two events to occur at the same time, namely when the King of Hearts is selected.

 

Independent Events—two events are independent if the occurrence of one does not affect the probability of the occurrence of the other; the way to check whether or not the events are independent is to check to see if

 

Dependent Events—two events are dependent if the occurrence of one affects the occurrence of the other.

 

Example: Consider the following two-way table.

 

                                                           

 

yes

no

total

MALE

15

45

60

FEMALE

4

36

40

TOTAL

19

81

100

 

Are the events “female” and “yes” independent?

 

Does

P(female) = 40/100 = 0.4

These events are dependent.  In terms of the problem, this means that the probability that someone says yes depends on whether you are asking a male or a female.

 

Example: Consider the following two-way table.

 

                                               

 

defective

good

total

MACHINE 1

9

51

60

MACHINE 2

6

34

40

TOTAL

15

85

100

 

Are the events “defective” and “Machine 1” independent?

 

Does

P(defective) = 15/100 = 0.15

These events are independent.  In terms of the problem, this means that the probability that a tape is defective does not depend on which machine produced the tape.

 

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5.2

 

Multiplication Rule for Independent Events—the probability of two independent events A and B occurring together:

                                                P(A and B) = P(A) * P(B)

 

Example: Let the experiment be flipping a coin and then rolling a die.  What is the probability of getting “heads” and “4?”  First, are these independent events?  Yes, they are because the occurrence of one does not affect the probability of the occurrence of the other.  Therefore, we can use the multiplication rule from above.

                                    P(heads and 4) = P(heads) * P(4)  = (1/2) * (1/6) = 1/12

 

Multiplication Rule for Dependent Events—P(A and B) = .

 

Example: A bag contains 4 green balls and 3 red balls.  If two balls are drawn from the bag, what is the probability that both are green?

 

Events: A = 1st ball is green; B = 2nd ball is green

Are A and B independent or dependent events?  Dependent

P(A and B) =  = 4/7 * 3/6 = 12/42 or 0.2857

 

Using the Multiplication Rule for Dependent Events to compute Conditional Probability:

 

Example: The probability that a randomly selected student from a college is a senior is 0.2.  The probability that a randomly selected student is a computer science major and a senior is 0.03.  Find the conditional probability that a randomly selected student is a computer science major given that he/she is a senior.

 

Solution: Two events: S = senior and C = computer science major

We know P(S) = 0.2 and P(S and C) = 0.03.  We are looking for

P(S and C) = P(S) *

So, 0.03 = 0.2x and x = 0.15.

 

Joint Probability of Mutually Exclusive Events—if A and B are mutually exclusive events, then P(A and B) = 0.

 

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5.3

 

Complementary Events—two mutually exclusive events that taken together include all outcomes for an experiment.

 

Example: Let the experiment be rolling a die.  Let A be the event that the number rolled is odd.  Then, the complement of A is the number rolled is even. 

 

Example: Let the experiment be selecting a card from a standard deck of 52 cards.  Let A be the event the card is a heart.  Then, the complement of A is the card is not a heart.

 

Note: P(A) + P(complement of A) = 1.

 

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Union of Events

 

Union of Events—the union of events A and B is the collection of all outcomes that belong either to A or to B or to both A and B; it is denoted “A or B” or “

 

Probability of the Union of Two Events-

 

                                    P(A or B) = P(A) + P(B) – P(A and B)

 

Example: The following table gives a 2-way classification of all faculty members at a local school.

 

                                                                                                           

                                                                       

           

tenured

non-tenured

total

MALE

74

28

102

FEMALE

29

12

41

TOTAL

103

40

143

 

One faculty member is selected at random.  Find each probability.

 

  • P(female or non-tenured)

            P(female or non-tenured) = P(female) + P(non-tenured) – P(female and non-    tenured)

            = (41/143) + (40/143) – (12/143)

            = (69/143)

            = 0.483

  • P(tenured or male)

      P(tenured or male) = P(tenured) + P(male) – P(tenured and male)

      = (103/143) + (102/143) – (74/143)

      = (131/143)

      = 0.916

 

Probability of the Union of Two Mutually Exclusive Events-

                                                P(A or B) = P(A) + P(B)

 

Example: The probability of a student getting an A in this course is 0.24 and that of getting a B is 0.28.  What is the probability that a randomly selected student from this class will get an A or a B?

P(A or B) = P(A) + P(B)

= 0.24 + 0.28

= 0.52

 

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Combinations

 

Combinations

  • The number of ways x elements can be selected from n elements.
  • The order in which the elements are chosen does not matter.
  • Denoted

     

                 

  • This is read “the number of combinations of n elements selected x at a time” or “n choose x.”
  • The formula for counting combinations is (# of outcomes if order mattered)/(# of ways to arrange x elements).

 

Example: Suppose that you wanted to choose 3 out of the first 5 letters of the alphabet.  How many combinations are possible?

 

  • If order mattered, how may outcomes would there be? 5*4*3 = 60 outcomes.
  • How many ways can you arrange a set of three letters? 3*2*1 = 6 ways.
  • Number of combinations possible: 60 divided by 6 = 10 combinations possible. 

 

Just so that you will understand what this number is telling you, here are the ten combinations of the first 5 letter taken 3 at a time.

 

ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE