Warm-UpWarm-Up

1.1. Expand (xExpand (x22 – 4) – 4)77

1.1. Find the 8Find the 8thth term of (2x + 3) term of (2x + 3)1010

14 12 10 8 6 4 228 336 2240 8960 21504 28672 16384x x x x x x x

32099520x

Probability, Odds and Probability, Odds and The Law of Large The Law of Large

NumbersNumbers

Basics of ProbabilityBasics of Probability

ExperimentExperiment – any observation of – any observation of random phenomenon.random phenomenon.

OutcomesOutcomes – the different possible – the different possible results of the experiment.results of the experiment.

Sample SpaceSample Space – The set of all – The set of all possible outcomes for an experiment.possible outcomes for an experiment.

Ex) What would be the sample space of Ex) What would be the sample space of 3 children being born to a family and 3 children being born to a family and we note the birth order with respect we note the birth order with respect to sex.to sex.

Find the sample spaceFind the sample space

We select 1 card from a standard We select 1 card from a standard deck of 52 and then without deck of 52 and then without returning the card, we select a returning the card, we select a second card.second card.

What is an event?What is an event?

An An EventEvent is a subset of the sample is a subset of the sample space.space.

The event of having 2 girls and 1 boy The event of having 2 girls and 1 boy would be a subset of the sample would be a subset of the sample space we found earlier. {bgg, gbg, space we found earlier. {bgg, gbg, ggb}ggb}

Empirical ProbabilityEmpirical Probability

The probability of an outcome in a sample The probability of an outcome in a sample space is a number between 0 and 1.space is a number between 0 and 1.

P(E) = the sum of the probabilities of the P(E) = the sum of the probabilities of the outcomes that make up E.outcomes that make up E.

To find the probability of an event ocurring To find the probability of an event ocurring P(E) = P(E) = the # of times E occurs_______the # of times E occurs_______

the # of times the experiment is performedthe # of times the experiment is performed

This ratio is the This ratio is the relative frequencyrelative frequency..

Empirical information from an Empirical information from an experimentexperiment

What is the What is the probability that the probability that the patient will develop patient will develop severe side effects?severe side effects?

What is the What is the probability that a probability that a patient receiving the patient receiving the flu vaccine will flu vaccine will experience no side experience no side effects?effects?

Side Side

EffectsEffects# of# of

TimesTimes

NoneNone 6767

MildMild 2525

SevereSevere 88

Theoretical information can also Theoretical information can also be used to determine probabilitybe used to determine probability

1.1. We flip 3 fair coins. What is the We flip 3 fair coins. What is the probability of each outcome in the probability of each outcome in the sample space?sample space?

1.1. We draw a 5-card hand randomly We draw a 5-card hand randomly from a standard 52-card deck. from a standard 52-card deck. What is the probability that we draw What is the probability that we draw one particular hand?one particular hand?

1

8

52 5 2,598,960 so the probability for each hand is

1

2,598,960

C

When each outcome in a sample When each outcome in a sample space is equally likely to occurspace is equally likely to occur

Then the probability of each outcome Then the probability of each outcome occurring isoccurring is

For an event E in the sample space: For an event E in the sample space:

1 1

# of outcomes in S ( )n S

( )( )

( )

n EP E

n S

What is the probability of:What is the probability of:

Rolling a total of 4 when rolling two Rolling a total of 4 when rolling two fair dice?fair dice?Using the FCP we know that there are 36 Using the FCP we know that there are 36

possible outcomes.possible outcomes.E = {(1,3), (2,2), (3,1)}E = {(1,3), (2,2), (3,1)}

( ) 3 1( )

( ) 36 12

n EP E

n S

What is the probability of:What is the probability of:

Drawing a 5-card hand and all 5 Drawing a 5-card hand and all 5 cards are hearts?cards are hearts?

Flipping 3 coins and getting 2 heads Flipping 3 coins and getting 2 heads and a tail in any order?and a tail in any order?

3/83/8

(13,5) 12870.000495

(52,5) 2598960

C

C

Probability & GeneticsProbability & Genetics 2 parents are 2 parents are

carriers of a disease carriers of a disease (d). Let (n) (d). Let (n) represent the normal represent the normal gene. Create a gene. Create a Punnett square to Punnett square to display the possible display the possible genes of a child.genes of a child.

What is the What is the probability that a probability that a child will have the child will have the disease?disease?

ParentParent

nn

11

dd

PP

a a nn

rr nnnn ndnd

ee

n n dd

tt

22

dndn dddd

OddsOdds

Odds are another way of stating the Odds are another way of stating the likelihood of an event. likelihood of an event.

Expressed as a ratioExpressed as a ratioSuccessful event: Not a successful Successful event: Not a successful

eventevent

Suppose 2 of 7 events are successful.Suppose 2 of 7 events are successful.Probability: 2/7 or 0.29Probability: 2/7 or 0.29Odds in favor: 2/5 or 2:5Odds in favor: 2/5 or 2:5Odd against: 5/2 or 5:2Odd against: 5/2 or 5:2

Mutually Exclusive EventsMutually Exclusive Events

When two events are mutually exclusive, When two events are mutually exclusive, you can add to find the probability that you can add to find the probability that either one occurs.either one occurs.

For mutually exclusive events A and B:For mutually exclusive events A and B:P(A or B) = P(A) + P(B)P(A or B) = P(A) + P(B)

P(A U B)P(A U B)

If not mutually exclusive then:If not mutually exclusive then:P(A U B) = P(A) + P(B) – P(A B)P(A U B) = P(A) + P(B) – P(A B)

Compound EventCompound EventEvent made up of two or more events that Event made up of two or more events that

can happen at the same time or one after can happen at the same time or one after the other.the other.

Events can be independent or dependentEvents can be independent or dependent Independent events – one event does not Independent events – one event does not

affect another event (rolling a die and affect another event (rolling a die and tossing a coin) tossing a coin) P(A and B)=P(A)*P(B)P(A and B)=P(A)*P(B)

Dependent events – one event affects Dependent events – one event affects another event (drawing two cards without another event (drawing two cards without replacement) replacement) P(A and B)=P(A)*P(B after A)P(A and B)=P(A)*P(B after A)

CardsCards Draw one card from a standard Draw one card from a standard

deckdeckWhich example is Which example is MUTALLY EXCLUSIVE MUTALLY EXCLUSIVE

EVENTS?EVENTS?

What is the probability that the card is What is the probability that the card is an ACE or a JACK? P(ace or jack)=?an ACE or a JACK? P(ace or jack)=?

What is the probability that the card is What is the probability that the card is a FACE card or a spade? P(face card or a FACE card or a spade? P(face card or spade)=?spade)=?

Who will win the prize?Who will win the prize?

A school has 45 K, 55 1A school has 45 K, 55 1stst graders, 60 2 graders, 60 2ndnd graders and 55 3graders and 55 3rdrd graders. The graders. The school has 15 teachers and 5 school has 15 teachers and 5 administrators.administrators.

A radio station is giving away prizes at A radio station is giving away prizes at the school. What is the probability the school. What is the probability that the winner is:that the winner is:

P(1P(1stst grader)=? grader)=? P(3P(3rdrd grade)=? grade)=?

P(teacher)=?P(teacher)=? P(K or adm)=?P(K or adm)=?

Suppose the events are NOT Suppose the events are NOT mutually exclusivemutually exclusive

P(jack or spade)?P(jack or spade)?

P(freshman or girl)?P(freshman or girl)?

P(even or less than 5)?P(even or less than 5)?

P(red or heart)? Draw one card from P(red or heart)? Draw one card from a standard decka standard deck

Are the events disjoint Are the events disjoint (mutually exclusive)?(mutually exclusive)?

P(A or B)=P(A) + P(B)P(A or B)=P(A) + P(B)Even or OddEven or OddK and 3K and 3rdrd grader grader

P(A or B)=P(A) + P(B) – P(A and B)P(A or B)=P(A) + P(B) – P(A and B)Red and heartRed and heartFace card and spadeFace card and spadeFreshman and girlFreshman and girlAdministrator and MaleAdministrator and Male

Find the probability:Find the probability:

P(jack or spade)? Draw one card from a P(jack or spade)? Draw one card from a standard deckstandard deck

P(freshman or girl)? 45 Freshmen, 30 girls P(freshman or girl)? 45 Freshmen, 30 girls and 15 boysand 15 boys

P(even or less than 5)? Roll die.P(even or less than 5)? Roll die.

P(red or heart)? Draw one card from a P(red or heart)? Draw one card from a standard deckstandard deck

Complementary EventsComplementary Events

Two events are complementary if they Two events are complementary if they are mutually exclusive and together are mutually exclusive and together they include all the possibilities.they include all the possibilities.Heart and Not a HeartHeart and Not a HeartRain and Not rainRain and Not rainSix and Not a SixSix and Not a Six

Example: Find P(not a face card) when a Example: Find P(not a face card) when a card is randomly chosen from a standard card is randomly chosen from a standard deck.deck.

The Law of Large Numbers The Law of Large Numbers (LLN)(LLN)

The LLN says that in The LLN says that in the long runthe long run relative frequency of relative frequency of repeated independent events gets closer and closer to the repeated independent events gets closer and closer to the true relative frequency as the number of trials increases.true relative frequency as the number of trials increases.

Note: The is not a Law of Averages. A common Note: The is not a Law of Averages. A common (mis)understanding of the LLN is that random phenomena (mis)understanding of the LLN is that random phenomena are suppose to compensate somehow for whatever has are suppose to compensate somehow for whatever has happened in the past. So, if results have fallen to one side happened in the past. So, if results have fallen to one side of what is expected are future results “due” in order to of what is expected are future results “due” in order to average out? (NO)average out? (NO)

For example:For example: A pregnant lady has five daughters. The probability that child A pregnant lady has five daughters. The probability that child

#6 is a girls is STILL 0.50 The sex of the baby is not #6 is a girls is STILL 0.50 The sex of the baby is not dependent on previous children.dependent on previous children.

You strike out 8 times in a row. Are you due a hit? The You strike out 8 times in a row. Are you due a hit? The probability that you will next time at bat is still 0.50 probability that you will next time at bat is still 0.50

Flip a coin 1 time and get 8 heads. The probability the next flip Flip a coin 1 time and get 8 heads. The probability the next flip is a tail is still 0.50 The coin does not know what has is a tail is still 0.50 The coin does not know what has happened in the past.happened in the past.

The LLN does…The LLN does…

promise that given a very large promise that given a very large number of trials (in the long run) the number of trials (in the long run) the distribution of subsequent results will distribution of subsequent results will eventually overwhelm any recent eventually overwhelm any recent drift away from what is expected.drift away from what is expected.

The long runThe long run is a long timeis a long time

LLNLLN

You flip a coin 5 times and get 5 heads.You flip a coin 5 times and get 5 heads.Suppose you continue to flip the coin Suppose you continue to flip the coin

100 times and end up with 54 heads 100 times and end up with 54 heads and 51 tails.and 51 tails.

The lesson of the LLN is that random The lesson of the LLN is that random processes do not need to compensate processes do not need to compensate in the short run to get back to the in the short run to get back to the right long-run probabilities.right long-run probabilities.

In the Long Run…In the Long Run…

If the probabilities do not change and If the probabilities do not change and the events are independent, the the events are independent, the probability of the next trial is always probability of the next trial is always the same, no matter what has the same, no matter what has happened up to then.happened up to then.

There is no Law of Averages for the There is no Law of Averages for the short term.short term.

Relative frequencies settle down in the Relative frequencies settle down in the long run and we then can officially give long run and we then can officially give the name “probability” to that value.the name “probability” to that value.

00<<probabilityprobability<<11

Roll one dieRoll one dieP(odd)=P(odd)=P(5 or 6)=P(5 or 6)=P(less than five)=P(less than five)=P(not greater than one)=P(not greater than one)=

Roll two diceRoll two diceP(3 and 5)=P(3 and 5)=P(sum less than 10)=P(sum less than 10)=P(two evens)=P(two evens)=

VocabularyVocabulary

Important to learn the terms and Important to learn the terms and definitions for this unit of study.definitions for this unit of study.

Read the book and study the Read the book and study the examples.examples.

Check HW answers with the back of Check HW answers with the back of the book.the book.

Warm-UpWarm-Up

1.1. Courtney is in charge of purchasing Courtney is in charge of purchasing the following items for Christmas the following items for Christmas Angels:Angels:3 stuffed animals, 2 action figures, 3 stuffed animals, 2 action figures, and 5 books. If she has a choice of and 5 books. If she has a choice of 10 stuffed animals, 15 action 10 stuffed animals, 15 action figures, and 20 books, how many figures, and 20 books, how many ways can she make her selections?ways can she make her selections?

2.2. Given a 10-sided die (decahedron):Given a 10-sided die (decahedron):a.a. What is the probability of rolling a What is the probability of rolling a

number greater than 2?number greater than 2?b.b. What are the odds of rolling a 7?What are the odds of rolling a 7?

195,350,400 ways

4/5

1:9

Given two 11-sided dice:Given two 11-sided dice: What is the probability of rolling a What is the probability of rolling a

sum of 7?sum of 7? What is the probability of rolling a What is the probability of rolling a

sum larger than 3?sum larger than 3?

Given a standard deck of 52 cards:Given a standard deck of 52 cards:5.5. What is the probability of drawing a What is the probability of drawing a

diamond or a face card?diamond or a face card?6.6. What is the probability of drawing 3 What is the probability of drawing 3

diamonds in a row? (no diamonds in a row? (no replacement) replacement)

6/121

118/121

11/26

11/850 0r 0.13