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CHAPTER 5 PROBABILITY

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CARDS & DICE BLACKRED CLUBSPADEDIAMONDHEARTTOTAL ACE11114 FACE CARD (K, Q, J) 333312 NUMBERED CARD (1-9) 999936 TOTAL13 52

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CARDS & DICE Single Die: Six sides Numbered 1 to 6 Double Dice: Use the sum of the two top sides (totals are 2 to 12)

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PROBABILITY DEFINITIONS: Probability – The likelihood that a given event will occur. Expressed as a proportion. Experiment – A procedure that can be repeated that has uncertain random results. Sample Space – All possible outcomes of an experiment or observation – Examples. Event – Subset of the Sample Space (could be simple or complex). Notation – P(A) = Probability of A occurring.

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TWO WAYS TO DETERMINE: Empirical – Trial; Run a number of times and count successes; Then Classical – Calculated based on known outcomes Examples of each.

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LAW OF LARGE NUMBERS As the number of trials increases using the Empirical Method, the closer the trial P(A) approaches the Classical (actual) P(A).

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PROBABILITY PROPERTIES The probability of an impossible event is 0.0. The probability of an absolutely certain event is 1.0. For any event A, 0.0 ≤P(A) ≤ 1.0. The sum of probabilities of all events in the Sample Space is 1.0.

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PROBABILITY DEFINITIONS: Probability Model: A table listing each outcome of the Sample Space and its Probability of Occurrence. For the Table to be a true Probability model, all outcomes must be listed; all the probabilities must meet probability definition; and the sum of all probabilites must add to 1.0.

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PROBABILITY DEFINITIONS: Unusual Event: Any event whose probability is less than 0.05 or 5%.

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PROBABILITY EXAMPLES For each, show sample space as well as probabilities: Coins Jar of Colored Marbles Triplets Cards Dice – Do full Probability Space

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DEFINE DISJOINT (5.2): Disjoint (Mutually Exclusive) Events – Events that can not occur simultaneous. Venn Diagram Examples of Mutually Exclusive Events Flipping a coin. Drawing a card. Picking a marble from a jar.

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ADDITION (OR) RULE: If Events A & B are Mutually Exclusive (or disjoint), the Probability of Event A or Event B occurring is P(A) + P(B).

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DEFINE NOT DISJOINT (Not Mutually Exclusive): Examples of events that are Not Disjoint: Cards of number and suit. Marbles of colors and letters. Venn Diagram

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ADDITION (OR) RULE: If Events A & B are NOT Mutually Exclusive (or disjoint), the Probability of Event A or Event B occurring is P(A) + P(B) – the overlap P(A and B). AND Venn Diagram Use with given probabilities.

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ADDITION (OR) RULE: Examples of disjoint probabilities Cards Marbles Tables

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ADDITION (OR ) RULE: FRESHMENSOPHOMOREJUNIORSENIORTOTAL SATISFIED57496461231 NEUTRAL2315161165 NOT SATISFIED2118142679 TOTAL101829498375

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COMPLIMENT RULE: Probability of A not occurring is

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COMPLIMENT RULE: If A, B, C, D & E are all possible events, then And

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DEFINE INDEPENDENCE(5.3): Two events are independent if the occurrence of one event does not affect the probability of the occurrence of the other event. In other words, P(B) is the same whether or not event A has occurred or not. Occurrence of event A does not affect probability of event B.

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MULTIPLICATION (AND) RULE: If two events are independent then Examples: Die Marbles with replacement. Games

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CONDITIONAL PROBABILITY(5.4): Means the probability of B occurring GIVEN THAT event A has already occurred. The statement “Given That” CHANGES THE SAMPLE SPACE. FORMULA: Do with given probabilities.

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CONDITIONAL PROBABILITY: FRESHMENSOPHOMOREJUNIORSENIORTOTAL SATISFIED57496461231 NEUTRAL2315161165 NOT SATISFIED2118142679 TOTAL101829498375

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MULTIPLICATION RULE FOR EVENTS THAT ARE NOT INDEPENDENT: If events A & B are NOT independent Examples Marbles without replacement Cards without replacement Table

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MULTIPLICATION RULE FOR LARGE POPULATION: If the population is very large (size of a large town) then we can consider two events as independent even without replacement (consider it as with replacement). Example of survey.

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ACCEPTANCE SAMPLING EXAMPLE: Lot of 100 circuits has 5 defective. Take 2 circuits without replacement. If only one defective then reject the lot. What is the probability of rejecting the lot? Build tree of good/bad using conditional probability. Use the addition & multiplication rules to find probability of lot being rejected.

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ACCEPTANCE SAMPLING EXAMPLE: Pass 94/99 2 nd Sample 95/100 Pass Fail 5/99 1 st Sample 5/100 Fail Pass 95/99 2 nd Sample Fail 4/99

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PROBABILITY SUMMARY Good Summary of Probability Rules on Page 284 of text. There is NO relationship between Mutually Exclusive and Independence concepts.

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COUNTING METHODS: Using the classical method of calculating probabilities, we need to find better ways to count possibilities. Example of births of triplets.

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COUNTING METHODS: Multiplication Rule with Replacement: How many three or five digit numbers? How many “word” combinations with 6 letters? How many meals? Formula: n items to select from and want to select r items -

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COUNTING METHODS: Multiplication Rule without Replacement: Define Factorial. How many three digit numbers using only numbers 0, 1, 2 without replacement? How many word combinations with 6 letters without replacement? How many ways to arrange 5 books without replacement?

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COUNTING METHODS: Multiplication Rule without Replacement and with more items than options: Have n items but only r places for them. Order matters: PERMUTATION 8 books but only 5 places 10 people 5 offices Formula:

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COUNTING METHODS: Multiplication Rule without Replacement and with more items than options: Have n items but only r places for them. Order does not matter: COMBINATION 8 books but only 5 places 10 people committee of 5 Lottery Formula:

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COUNTING METHODS: CALCULATOR FUNCTIONS

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COUNTING METHODS: Arrange n items n ways, if not all n items are distinct. Formula: Where the are non distinct items. Example using fruits or names.

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COUNTING METHODS: SUMMARY ON PAGE 304 READ SECTION 5.6

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PROBABILITY QUOTES “The 50-50-90 rule: Anytime you have a 50-50 chance of getting something right, there's a 90% probability you'll get it wrong.” Andy Rooney “The 50-50-90 rule: Anytime you have a 50-50 chance of getting something right, there's a 90% probability you'll get it wrong.” Andy Rooney “From principles is derived probability, but truth or certainty is obtained only from facts.” Tom Stoppard “From principles is derived probability, but truth or certainty is obtained only from facts.” Tom Stoppard “Life is a school of probability.” Walter Bagehot “Life is a school of probability.” Walter Bagehot

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PROBABILITY

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