Chapter 4 Elementary Probability Theory

Chapter 4 Elementary Probability TheoryWhat is Probability?Probability is a numerical measure between 0 and 1 that describes the likelihood that an event will occur. Probabilities closer to 1 indicate that the event is more likely to occur. Probabilities closer to 0 indicated that the event is less likely to occur.Note:P(A) = probability of event A; you read it as P of A.

P(A)=1, the event A is certain to occur

P(A)=0, the event A is certain to not occur

Binary number works like this1 means its true, 0 means false.See if you understand this statement:There are only 10 types of people in the world: those who understand binary, and those who don'tAnywaysProbability AssignmentsExamplesIntuition NBA announcer claims that Kobe makes 84% of his free throws. Based on this, he will have a high chance of making his next free throw.

Relative frequency Auto Fix claims that the probability of Toyota breaking down is .10 based on a sample of 500 Toyota of which 50 broke down.

Equally likely outcome - You figure that if you guess on a SAT test, the probability of getting it right is .20

Group WorkCreate a situation for each of the probability assignments. (intuition, relative frequency, equally likely outcome)

Show meLaw of Large NumbersIn the long run, as the sample size increases, the relative frequencies of outcomes get closer to the theoretical (or actual) probability value

Example: The more numbers you ask, the more likelihood that P(getting a girls real number)=1Law of Large Numbers examples:The more numbers you ask, the more likelihood that P(getting a (hot) girls real number)=1

Then after collecting all the numbers, the more girls you ask out on a date, the more likelihood that P(getting a date)=1

Some other real life examples:Casino (the more you play, the more you lose)Insurance (the more people you insure, the less the likelihood the company have to pay for the insurance benefits)

Statistical ExperimentStatistical experiment or statistical observation can be thought of as any random activity that results in a definite outcomeAn event is a collection of one or more outcomes of a statistical experiment or observationSimple event is one particular outcome of a statistical experimentThe set of all simple events constitutes the sample space of an experimentExample: Blue eyes vs Brown eyes (relating to biology)Brown eyes genotype is Bb or BBBlue eyes genotype is bb

If your Dad has Brown eyes (and his dad has blue eyes) and your Mom has blue eyes, whats the probability that you have blue eyes?Answer (using sample space)BbbBbbbbBbbbDadMomGroup Work (use sample space):You are running out of time in a true/false quiz. You only have 4 questions left! How should you guess?

P(all false)= P(3 false)=

P(all true)= P(2 false)=

P(1 true)= P(1 false)=

P(2 true)=

P(3 true)=AnswerYour sample space should have 16 different combinations

P(all false)= 1/16 P(3 false)= 4/16

P(all true)= 1/16 P(2 false)= 6/16

P(1 true)= 4/16 P(1 false)= 4/16

P(2 true)= 6/16

P(3 true)= 4/16

You will probably choose 2 true and 2 false

TTTTFTTTTFTTTTFTTTTFFFTTFTFTFTTFTFFTTFTFTTFFFFFTTFFFFTFFFFTFFFFFNote:Example:P(getting A in Mr. Lius class)+P(not getting A in Mr. Lius class) =1

P(getting A in Mr. Lius class)=.15

Whats the P(not getting A in Mr. Lius class)?AnswerP(not getting A in Mr. Lius class)= .85Group WorkP(having a date on a Friday)=1/7

Whats the P(not having a date on a Friday)?Answer6/7Homework Practice:Pg 130 #1-6 (all), 7-13 (odd)Compound EventsConsider these two situationP(5 on 1st die and 5 on 2nd die)

P(ace on 1st card and ace on 2nd card)

What is the difference between these two situation?The answerIn the first situation, the first result does not effect the outcome of the 2nd result.

In the second situation, the first result does effect the outcome of the 2nd result.IndependentTwo events are independent if the occurrence or nonoccurrence of one does not change the probability that the other will occur

What does it mean if two events are dependent?Multiplication rule for independent eventsWhat if the events are dependent?Then we must take into account the changes in the probability of one event caused by the occurrence of the other event.Sample SpaceABA and BGeneral multiplication rule for any eventsConditional Probability Example:Your friend has 2 children. You learned that she has a boy named Rick. What is the probability that Ricks sibling is a boy?

Take a guess AnswerGroup WorkA machine produce parts thats either good (90%), slightly defective (2%) or obliviously broken (8%). The parts gets through an automatic inspection machine that is able to find the oblivious broken parts and throw them away. What is the probability of the quality part that make it through and get shipped?AnswerRelax!Conditional Probability can be very intriguing and complicated. We wont go into any more in depth..or maybe.Note:Very important to understand about probability is that are the events dependent or independent.Group WorkSuppose you are going to throw 2 fair dice. What is the probability of getting a 3 on each die?

A) Is this situation independent or dependent?B) Create all the sample space (all the potential outcomes)C) What is the probability?AnswerA) Independent because one event does not affect the second eventB) You should have 36 total outcomesC) 1/36Group WorkI took a die away. Now you only have ONE die! Again you toss the die twice. What is the probability of getting a 1 on the first and 4 on the second try?AnswerIt is still an independent event!1/36Note:The last two examples are considered multiplication rule, independent events.

Group WorkMr. Liu has a 80% probability of teaching statistics next year. Mr. Riley has a 15% probability of teaching statistics next year. What is the probability that both Mr. Liu and Mr. Riley teach statistics next year?Answer.8*.15=.12 or 12% probabilityNow comes the dependent eventsSuppose you have 100 Iphones. The defective rate of iphone is 10%. What is the probability that you choose two iphones and both are defective?AnswerGroup workWhat is the probability of getting tail and getting a 3 on a die and getting an ace in a deck of cards?AnswerAddition RulesYou use addition when you want to consider the possibility of one event OR another occurring Example:Group Work: And or Or?1) Satisfying the humanities requirement by taking a course in the history of Japan or by taking a course in classical literature2) Buying new tires and aligning the tires3) Getting an A in math but also in biology4) Having at least one of these pets: cat, dog, bird, rabbitAnswer1) or2) and3) and4) orNote:Two events are mutually exclusive or disjoint if they cannot occur together. In particular, events A and B are mutually exclusive if P(A and B)=0

Addition rule for mutually exclusive events A and BP(A or B)=P(A)+P(B)

General Rule for any events A and BP(A or B)=P(A)+P(B)-P(A and B)

Remember in mutually exclusive events P(A and B)=0Group Example:Employee typeDemocrat (D)Republican (R)Independent (I)Row TotalExecutive (E)534948Production Worker (PW)6321892Column Total685517140 grand totalAnswerHomework Practice:Pg 146 #1,2,5,7,9,10,14,19Conditional Probability extensionBayess theorem: It uses conditional probabilities to adjust calculations so that we can accommodate new relevant information.

The special case where event B is partitioned into only two mutually exclusive events.FormulaExample (Real Life Extension): How accurate is one pregnancy test?Supposedly pregnancy test strip claims it is 99% accurate.

false-positive and false-negative

NOT PregnantPregnant!!!!Pregnancy Test NegativeTrue NegativeFalse NegativePregnancy Test PositiveFalse PositiveTrue Positive!!ProcedureTrees and Counting TechniquesTree diagram:A tree diagram shows all the possible outcomes of an event.All possible outcomes of an event are shown by a tree diagram.

Example using tree diagrams:If a coin and a dice are tossed simultaneously, what is the probability of getting tail and even number?

Answer1/4Group Work using tree diagram:You are on a sports team. What is the probability that out of three games, you win two of them?Tree Diagram with Probability:You have 7 balls, 4 are blue and 3 are green. What is the probability that when you pick the balls, you get green on 1st and blue one 2nd?Shown in classGroup Work:You make free throws 85% of the time. What is the probability of making at least one out of the three?P(make 1 out of 3)=99.66% of the timeFactorials! Means factorial

0!=11!=1n!=(n)(n-1)(n-2)(n-3).What is 6! ?6!=6*5*4*3*2*1=720Combination vs PermutationCombination:Order does not matter! It is not importantIf you have 3,1,2, it is the same as 1,3,2 because they all have 1,2,3

Different Arrangement of things

Combination is choosingPermutation:Order does matter! It is important

Note: Permutation is positionCombination vs Permutation continue:Both of them break down into two different category:

Combination with repetition (ex: ice cream scoops)Combination without repetition (ex: lottery)

Permutation with repetition (ex: lock in locker room) Permutation without repetition (ex: marathon race)Reading activity:http://www.mathsisfun.com/combinatorics/combinations-permutations.html

Read the different examplesWhew that was a lot of readingnow do some examples.You have 8 people, what are the number of possible ordered seating arrangement for 5 chairsAnswerGroup Work:Gamestop has 25 new games this month and you decided to buy 5 of them. How many different arrange of game you can have?AnswerGroup Work: MEGA MillionWhat is the chance of winning the jackpot for MEGA Million?

You have 5 slots + 1 slot for MEGA number

The first 5 slots are numbers between 1-75, mega number is number between 1-15AnswerGroup Work: Powerball lotteryWhat is the chance of winning the Jackpot for Powerball?

You have 5 slots+1 slot for Power number

The first 5 slots are numbers between 1-56, powerball slot is number between 1-35Answer1 in 175,223,510Interesting factEven though its harder to win the Jackpot, for overall winning chance, you have more chance for MEGA million than PowerballHomework PracticePg 160 #1-27 every other odd