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Chapter 5 Probability Models

Constructing Models of Random Behavior Section 5.1 Constructing Models of Random Behavior

Fundamental Probability Facts Event: possible outcome from a random situation

Fundamental Probability Facts Event: possible outcome from a random situation Flip a coin: getting heads Rolling a standard die: rolling a 6 Drawing a card from a standard deck: getting a face card; getting a black ace

Mutually exclusive events: AKA disjoint events.

Events that can not occur on the same opportunity Mutually exclusive events: AKA disjoint events. Events that can not occur on the same opportunity

Mutually Exclusive

Not Mutually Exclusive

Fundamental Probability Facts Sample space: a complete list or description of disjoint (mutually exclusive) outcomes of a chance process

Sample Space Examples Flip a coin: H T

Flip two coins: Possible outcomes Sample Space Examples Flip two coins: Possible outcomes H HH H T HT T H TH T TT

Sample Space Examples Roll a standard die: 1 2 3 4 5 6

Flip a coin, then roll a die Sample Space Examples Flip a coin, then roll a die 1 H1 2 H2 H 3 H3 4 H4 5 H5 6 H6 1 T1 2 T2 T 3 T3 4 T4 5 T5 6 T6

Probability can be expressed as a fraction, decimal, or percent

Probability can be expressed as a fraction, decimal, or percent Ranges from 0 to 1 or 0% to 100%

Probability of an event P(E) =

Probability of an event P(E) = If we flip a coin: P(heads) =

P(events that can never happen) = ? P(events that are certain to happen) = ?

P(events that can never happen) = 0 P(events that are certain to happen) =

P(events that can never happen) = 0 P(events that are certain to happen) = 1

The Law of Large Numbers In random sampling,

The Law of Large Numbers In random sampling, the larger the sample, the closer the proportion of successes in the sample tends to be to the proportion of successes in the population.

The Law of Large Numbers In random sampling, the larger the sample, the closer the proportion of successes in the sample tends to be to the proportion in the population. In other words, the more trials you conduct, the closer you can expect your experimental probability to be to the theoretical probability

The Law of Large Numbers Spinning a Penny

Rolling a Die When you roll a die, (a) what are the outcomes (b) what is the theoretical probability of each outcome?

Rolling a Die When you roll a die, (a) what are the outcomes -- 1, 2, 3, 4, 5, or 6 (b) what is the theoretical probability of each outcome?

Rolling a Die When you roll a die, (a) what are the outcomes -- 1, 2, 3, 4, 5, or 6 (b) what is the theoretical probability of each outcome?

Rolling a Die

A student playing monopoly says “I have not rolled doubles on the last six rolls; I am due for doubles.” How does the Law of Large Numbers apply here?

How does the Law of Large Numbers apply here? A student playing monopoly says “I have not rolled doubles on the last six rolls; I am due for doubles.” The dice will eventually come up doubles, but the probability remains on each roll no matter what has happened before. Any one particular random trial is just that - - random.

Law of Large Numbers In a random process: can not predict accurately what happens in an individual trial or even in a small number of trials

Law of Large Numbers In a random process: can not predict accurately what happens in an individual trial or even in a small number of trials but you can predict the pattern that will emerge if the process is repeated a large number of times.

Fundamental Principle of Counting Tree diagram: Shows all possible outcomes of an experiment Quickly becomes unwieldy if many stages or many outcomes for stages.

Fundamental Principle of Counting Tree diagram: Shows all possible outcomes of an experiment Quickly becomes unwieldy if many stages or many outcomes for stages. Think about drawing a card from a standard deck of playing cards, replacing it, then repeating this process two more times

Fundamental Principle of Counting If you only need to know how many outcomes are possible, then use the Fundamental Principle of Counting.

Fundamental Principle of Counting For a two-stage process with n1 possible outcomes for stage 1 and n2 possible outcomes for stage 2, the number of total possible outcomes for the two stages is n1 n2. This can be extended to as many stages as desired.

Fundamental Principle of Counting How many outcomes are possible if you flip a coin, roll a die, and pick a card from a standard deck of playing cards?

Fundamental Principle of Counting How many outcomes are possible if you flip a coin, roll a die, and pick a card from a standard deck of playing cards? coin die card 2 ● 6 ● 52 = 624 outcomes

Fundamental Principle of Counting Suppose you flip a fair coin seven times. a) How many possible outcomes are there? b) What is the probability you will get seven heads? c) What is the probability that you will get heads six times and tails once?

Fundamental Principle of Counting Suppose you flip a fair coin seven times. a) How many possible outcomes are there? b) What is the probability you will get seven heads? c) What is the probability that you will get heads six times and tails once?

Fundamental Principle of Counting Suppose you flip a fair coin seven times. a) How many possible outcomes are there? b) What is the probability you will get seven heads? c) What is the probability that you will get heads six times and tails once?

Fundamental Principle of Counting Suppose you flip a fair coin seven times. a) How many possible outcomes are there? b) What is the probability you will get seven heads? c) What is the probability that you will get heads six times and tails once?

Two-Way Table When a process has only two stages, it is often more convenient to list them using a two-way table.

Two-Way Table When a process has only two stages, it is often more convenient to list them using a two-way table. Make a two-way table that shows all possible outcomes when you roll two fair dice.

Two-Way Table Second Roll 1 2 3 4 5 6 1 First Roll 2 3 4 5 6

Make a two-way table that shows all possible outcomes when you roll two fair dice. Page 295

Make a table that gives the probability distribution for the sum of the two dice. The first column should list the possible sums, and the second column should list their probabilities. Sum Probability

Fundamental Principle of Counting Suppose you ask a person to taste a particular brand of strawberry ice cream and evaluate it as good, okay, or poor on flavor and as acceptable or unacceptable on price. How many possible outcomes are there? Show all possible outcomes on a tree diagram. Are all the outcomes equally likely?

Fundamental Principle of Counting 6 outcomes (3 flavor choices times 2 price choices)

Fundamental Principle of Counting Suppose you ask a person to taste a particular brand of strawberry ice cream and evaluate it as good, okay, or poor on flavor and as acceptable or unacceptable on price. How many possible outcomes are there? Show all possible outcomes on a tree diagram. Are all the outcomes equally likely?

Fundamental Principle of Counting b)

Fundamental Principle of Counting Suppose you ask a person to taste a particular brand of strawberry ice cream and evaluate it as good, okay, or poor on flavor and as acceptable or unacceptable on price. How many possible outcomes are there? Show all possible outcomes on a tree diagram. Are all the outcomes equally likely?

Fundamental Principle of Counting It is impossible to tell the probabilities without knowing the quality of the ice cream and price. But it is unlikely that the outcomes are equally likely.

Page 297, P2

Page 297, P2

Page 297, P2

Page 297, P2 c. There appears to be a bias toward predicting a temperature that is too warm. When the prediction was incorrect, the actual temperature was colder than predicted ten times but was warmer only four times.

Page 298, P7

Page 298, P7

Page 298, P7 (b) about 0.44

Questions? Bring a penny to class tomorrow