statistics. probability experiment: an action through which specific results (counts, measurements,...

2.1 BASIC CONCEPTS OF PROBABILITY

Statistics

DEFINITIONS Probability experiment: An action through which specific

results (counts, measurements, or responses) are obtained.

Outcome: The result of a single trial in a probability experiment.

Sample Space: The set of all possible outcomes of a probability experiment

Event: One or more outcomes and is a subset of the sample space

HINTS TO REMEMBER: Probability experiment: Roll a six-sided

die

Sample space: {1, 2, 3, 4, 5, 6}

Event: Roll an even number (2, 4, 6)

Outcome: Roll a 2, {2}

IDENTIFYING SAMPLE SPACE OF A PROBABILITY EXPERIMENT

A probability experiment consists of tossing a coin and then rolling a six-sided die. Describe the sample space.

There are two possible outcomes when tossing a coin—heads or tails. For each of these there are six possible outcomes when rolling a die: 1, 2, 3, 4, 5, and 6. One way to list outcomes for actions occurring in a sequence is to use a tree diagram. From this, you can see the sample space has 12 outcomes.

TREE DIAGRAM FOR COIN AND DIE EXPERIMENT

{H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}

|

H1 T6T5T4T3T2T1H6H5H4H3H2

1

654321654321

HT

SOME MORE

A probability experiment consists of recording a response to the survey statement below and tossing a coin. Identify the sample space.

Survey: There should be a limit to the number of terms a US senator can serve.

Agree Disagree No opinion

SOME MORE A probability experiment consists of

recording a response to the survey statement below and tossing a coin. Identify the sample space.

A. Start a tree diagram by forming a branch for each possible response to the survey.




B. At the end of each survey response branch, draw a new branch for each possible coin outcome.


H H HT T T



C. Find the number of outcomes in the sample space. In this case -- 6Agree Disagree No opinion

H H HT T T



D. List the sample space {Ah, At, Dh, Dt, Nh, Nt}Agree Disagree No opinion

H H HT T T

SIMPLE EVENTS Simple Event: An event that consists of a single outcome. Decide whether the event is simple or not. Explain your

reasoning: 1. For quality control, you randomly select a computer chip

from a batch that has been manufactured that day. Event A is selecting a specific defective chip. (Simple because it has only one outcome: choosing a specific defective chip. So, the event is a simple event.

2. You roll a six-sided die. Event B is rolling at least a 4. B has 3 outcomes: rolling a 4, 5 or 6. Because the event has

more than one outcome, it is not simple.

SIMPLE EVENTS You ask for a student’s age at his or her last

birthday. Decide whether each event is simple or not:

1. Event C: The student’s age is between 18 and 23, inclusive.

A. Decide how many outcomes are in the event.

B. State whether the event is simple or not.

SIMPLE EVENTS You ask for a student’s age at his or her last birthday. Decide

whether each event is simple or not:

1. Event C: The student’s age is between 18 and 23, inclusive.

A. Decide how many outcomes are in the event. The student’s age can be {18, 19, 20, 21, 22, or 23} = 6 outcomes

B. State whether the event is simple or not. Because there are 6 outcomes, it is not a simple event.



1. Event D: The student’s age is 20.

A. Decide how many outcomes are in the event.

B. State whether the event is simple or not.



1. Event D: The student’s age is 20.

A. Decide how many outcomes are in the event. There is only 1 outcome – the student is 20.

B. State whether the event is simple or not. Since there is only one outcome, it is a simple event.

TYPES OF PROBABILITY Three types of probability:

1. Classical probability 2. empirical probability 3. subjective probability

CLASSICAL PROBABILITY

P(E) = # of outcomes in E___________

Total # of outcomes in sample space

When P(E)=1 we say an event is certain When P(E)=0 we say an event is impossible The sum of all probabilities in the sample

space is 1

P( )- Probability functionE- Event

CLASSICAL PROBABILITY You roll a six-sided die. Find the

probability of the following: 1. Event A: rolling a 3 2. Event B: rolling a 7 3. Event C: rolling a number less than

5.


P(E) = # of outcomes in E___________

Total # of outcomes in sample space

You roll a six-sided die. Find the probability of the following:

First when rolling a six-sided die, the sample space consists of six outcomes {1, 2, 3, 4, 5, 6}


P(E) = # of outcomes in E___________ Total # of outcomes in sample space


1. Event A: rolling a 3 There is one outcome in event A = {3}. So,

P(3) = 1/6 = 0.167




2. Event B: rolling a 7 Because 7 is not in the sample space, there are no outcomes in event B. So,

P(7) = 0/6 = 0




3. Event C: rolling a number less than 5. There are four outcomes in event C {1, 2, 3, 4}. So

P(number less than 5) = 4/6 = 2/3 ≈0.667

EXAMPLE You select a card from a standard deck. Find the

probability of the following: 1. Event D: Selecting a seven of diamonds. 2. Event E: Selecting a diamond 3. Event F: Selecting a diamond, heart, club or

spade.

A. Identify the total number of outcomes in the sample space.

B. Find the number of outcomes in the event. C. Use the classical probability formula.


probability of the following: 1. Event D: Selecting a seven of diamonds. A. Identify the total number of outcomes in the

sample space. (52) B. Find the number of outcomes in the event.

(1) C. Use the classical probability formula.

P(7 Diamond) = 1/52 or 0.0192


probability of the following: 2. Event E: Selecting a diamond

A. Identify the total number of outcomes in the sample space. (52)

B. Find the number of outcomes in the event. (13) C. Use the classical probability formula.

P(Diamond) = 13/52 or 0.25


probability of the following: 3. Event F: Selecting a diamond, heart, club or

spade.

A. Identify the total number of outcomes in the sample space. (52)

B. Find the number of outcomes in the event. (52) C. Use the classical probability formula.

P(Any suite) = 52/52 or 1.0

CONDITIONAL PROBABILITY Is the probability of an event occurring

given that another event has already occurred. The conditional probability of event B occurring, given that event A has occurred, is denoted by P(B|A) and is read as “probability of B, given A.”

FINDING CONDITIONAL PROBABILITIES

Gene Present

Gene not present

Total

HighIQ

33 19 52

Normal IQ

39 11 50

Total 72 30 102

The table shows the results of a study in which researchers examined a child’s IQ and the presence of a specific gene in the child. Find the probability that the child has a high IQ, given that the child has the gene.Solution: There are

72 children who have the gene. So, the sample space consists of these 72 children. Of these, 33 have high IQ. So, P(B|A) = 33/72 ≈ .458

CONDITIONAL PROBABILITY & THE MULTIPLICATION RULE

Statistics

CLASSIFYING EVENTS AS INDEPENDENT OR DEPENDENT Decide whether the events are independent

or dependent.

3. Practicing the piano (A), and then becoming a concert pianist (B).

Solution: If you practice the piano, the chances of becoming a concert pianist are greatly increased, so these events are dependent.

INDEPENDENT AND DEPENDENT EVENTS

The question of the interdependence of two or more events is important to researchers in fields such as marketing, medicine, and psychology. You can use conditional probabilities to determine whether events are independent or dependent.

DEFINITION Two events are independent if the occurrence

of one of the events does NOT affect the probability of the occurrence of the other event. Two events A and B are independent if:

P(B|A) = P(B) or if P(A|B) = P(A)

Events that are not independent are dependent.

This reads as the probability of B given A


or dependent.

1. Selecting a king from a standard deck (A), not replacing it and then selecting a queen from the deck (B).

Solution: P(B|A) = 4/51 and P(B) = 4/52. The occurrence of A changes the probability of the occurrence of B, so the events are dependent.


or dependent.

2. Tossing a coin and getting a head (A), and then rolling a six-sided die and obtaining a 6 (B).

Solution: P(B|A) = 1/6 and P(B) = 1/6. The occurrence of A does not change the probability of the occurrence of B, so the events are independent.

THE MULTIPLICATION RULE To find the probability of two events

occurring in a sequence, you can use the multiplication rule.

The probability that two events A and B will occur in sequence is

P(A ∩ B) = P(A) ● P(B)- For events that are independent

P(A ∩ B) = P(A) ●P(B|A)- For events that are dependent∩ means “and”

USING THE MULTIPLICATION RULE TO FIND PROBABILITIES

Two cards are selected without replacement, from a standard deck. Find the probability of selecting a king and then selecting a queen.

Solution: Because the first card is not replaced, the events are dependent.

P(K and Q) = P(K) ● P(Q|K)006.0

265216

514

524


A coin is tossed and a die is rolled. Find the probability of getting a head and then rolling a 6.

Solution: The events are independent

P(H and 6) = P(H) ● P(6)

So the probability of tossing a head and then rolling a 6 is about .0083

083.0121

61

21


A coin is tossed and a die is rolled. Find the probability of getting a head and then rolling a 2.

P(H) = ½. Whether or not the coin is a head, P(2) = 1/6—The events are independent.

083.0121

61

21)2()()2( PHPHandP

So, the probability of tossing a head and then rolling a two is about .083.


The probability that a salmon swims successfully through a dam is .85. Find the probability that 3 salmon swim successfully through the dam. The probability that each salmon is successful is .85. One salmon’s chance of success is independent of the others.

614.)85)(.85)(.85(.)3( uccessfulsalmonaresP

So, the probability that all 3 are successful is about .614.


Find the probability that none of the salmon is successful.

003.)15)(.15)(.15(.)3( ulOTsuccessfsalmonareNP

So, the probability that none of the 3 are successful is about .003.


Find the probability that at least one of the salmon is successful in swimming through the dam.

997.003.1(1)1(

cessfulNonearesucPulissuccessfatleastP

So, the probability that at least one of the 3 are successful is about .997.

THE ADDITION RULE

A card is drawn from a deck. Find the probability the card is a king or a 10.

A = the card is a king B = the card is a 10.

P(A B) = P(A) + P(B) – P(A∩B)

is the symbol for “or”

MUTUALLY EXCLUSIVE EVENTS When event A and event B can not

happen at the same time This means that P(A∩B) = 0 The Addition Rule is simplified to

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

EMPIRICAL PROBABILITY

Based on observation obtained from probability experiments. The empirical probability of an event E is the relative frequency of event E

nf

encyTotalfrequfeventEFrequencyoEP )(

EXAMPLE

A pond containing 3 types of fish: bluegills, redgills, and crappies. Each fish in the pond is equally likely to be caught. You catch 40 fish and record the type. Each time, you release the fish back in to the pond. The following frequency distribution shows your results. Fish Type # of Times Caught, (f)Bluegill 13Redgill 17Crappy 10

f = 40

If you catch another fish, what is the probability that it is a bluegill?

EMPIRICAL PROBABILITY EX. CONTINUED The event is “catching a bluegill.” In

your experiment, the frequency of this event is 13. because the total of the frequencies is 40, the empirical probability of catching a bluegill is:

P(bluegill) = 13/40 or 0.325

EXAMPLE An insurance company determines that in every

100 claims, 4 are fraudulent. What is the probability that the next claim the company processes is fraudulent.

A. Identify the event. Find the frequency of the event. (finding the fraudulent claims, 4)

B. Find the total frequency for the experiment. (100)

C. Find the relative frequency of the event. P(fraudulent claim) = 4/100 or .04

LAW OF LARGE NUMBERS As you increase the number of times a

probability experiment is repeated, the empirical probability (relative frequency) of an event approaches the theoretical probability of the event. This is known as the law of large numbers.

USING FREQUENCY DISTRIBUTIONS TO FIND PROBABILITY

You survey a sample of 1000 employees at a company and record the ages of each. The results are shown below. If you randomly select another employee, what is the probability that the employee is between 25 and 34 years old?

Employee Ages Frequency

Age 15-24 54

Age 25-34 366

Age 35-44 233

Age 45-54 180

Age 55-64 125

65 and over 42

f = 1000

USING FREQUENCY DISTRIBUTIONS TO FIND PROBABILITY

P(age 25-34) = 366/1000 = 0.366

Employee Ages Frequency

Age 15-24 54

Age 25-34 366

Age 35-44 233

Age 45-54 180

Age 55-64 125

65 and over 42

f = 1000

EXAMPLE

Find the probability that an employee chosen at random is between 15 and 24 years old.

P(age 15-24) = 54/1000 = .054Employee Ages Frequency

Age 15-24 54

Age 25-34 366

Age 35-44 233

Age 45-54 180

Age 55-64 125

65 and over 42

f = 1000

SUBJECTIVE PROBABILITY

Subjective probability result from intuition, educated guesses, and estimates. For instance, given a patient’s health and extent of injuries, a doctor may feel a patient has 90% chance of a full recovery. A business analyst may predict that the chance of the employees of a certain company going on strike is .25

A probability cannot be negative or greater than 1, So, the probability of event E is between 0 and 1, inclusive. That is 0 P(E) 1

CLASSIFYING TYPES OF PROBABILITY Classify each statement as an example of

classical probability, empirical probability or subjective probability. Explain your reasoning.

1. The probability of your phone ringing during the dinner hour is 0.5

This probability is most likely based on an educated guess. It is an example of subjective probability.



2. The probability that a voter chosen at random will vote Republican is 0.45.

This statement is most likely based on a survey of voters, so it is an example of empirical probability.



3. The probability of winning a 1000-ticket raffle with one ticket is 1/1000.

Because you know the number of outcomes and each is equally likely, this is an example of classical probability.



Based on previous counts, the probability of a salmon successfully passing through a dam on the Columbia River is 0.85.

Event: = salmon successfully passing through a dam on the Columbia River.

Experimentation, Empirical probability.

PROPERTIES OF PROBABILITY The sum of the probabilities of all outcomes in

a sample space is 1 or 100%. An important result of this fact is that if you know the probability of event E, you can find the probability of the complement of event E.

The complement of Event E, is the set of all outcomes in a sample space that are not included in event E. The complement of event E is denoted by E’ and is read as “E prime.”

EXAMPLE For instance, you roll a die and let E be the event “the number

is at least 5,” then the complement of E is the event “the number is less than 5.” In other words, E = {5, 6} and E’ = {1, 2, 3, 4}

Using the definition of the complement of an event and the fact that the sum of the probabilities of all outcomes is 1, you can determine the following formulas:

P(E) + P(E’) = 1 P(E) = 1 – P(E’) P(E’) = 1 – P(E)

FINDING THE PROBABILITY OF THE COMPLEMENT OF AN EVENT Use the frequency distribution given in example

5 to find the probability of randomly choosing an employee who is not between 25 and 34.

P(age 25-34) = 366/1000 = 0.366

So the probability that an employee is not between the ages of 25-34 is

P(age is not 25-34) = 1 – 366/1000 = 634/1000 = 0.634

EXAMPLE: USE THE FREQUENCY DISTRIBUTION IN EXAMPLE 4 TO FIND THE PROBABILITY THAT A FISH THAT IS CAUGHT IS NOT A REDGILL.

A. Find the probability that the fish is a redgill. 17/40 = .425

B. Subtract the resulting probability from 1— 1-.425 = .575

C. State the probability as a fraction and a decimal.

23/40 = .575

Fish Type # of Times Caught, (f)Bluegill 13Redgill 17Crappy 10

f = 40

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