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Probability Chapter 3

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Methods of Counting The type of counting important for probability theory involves choosing the number of ways we can arrange a set of items.

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Permutation Permutation- any ordered sequence of a group or set of things. Tractors in a showroom window Model 50 Model 60 Model 70

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One way to solve is to list and count all possible combinations of the tractors. 1.506070 2.507060 3.605070 4.607050 5.705060 6.706050

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We can use a tree diagram as well

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3 ways to do the first step. 2 ways to do the second step. 1 way to do the last step. (3)(2)(1) or six permutations This is the multiplication rule. As the number of steps increases the calculation may become quite involved.

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General Rule If we can perform the first step in N 1 ways the second step in N 2 ways and so on for r steps, then the total number of ways we can perform the r steps is given by their product. (n 1 )(n 2 )(n 3 )…(n r )

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Luncheon menu has 3 appetizers 5 main dishes 4 beverages 6 desserts (3)(5)(4)(6)= 360

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When every object in the set is included in the permutation the number of permutations is nPn=n! Example: Four farm workers Four different jobs 4 P 4 =4!(4)(3)(2)(1)= 24 possible comb.

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For other counting problems we are interested in the permutation of a subset r of the n objects. nPr=n!/n-r!

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2 sales people are to be selected and given outside sales jobs from the six sales people in the district office. The rest will remain inside. 6P 2 =6!/(6-2)!=6!/4!=30 (6)(5)(4)(3)(2)(1)72030 (4)(3)(2)(1)24

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Combinations don’t depend on order Group r objects together from a set of n nCr 4 letters combination of three wxyzwxy wxz wyz xyz

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Number of permutations/number of permutations per combination. nCr=n!/r!(n-r)! (4)(3)(2)(1)/(3)(2)(1) (4-3)! 24/6=4

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Probability- used when a conclusion is needed in a matter that has an uncertain outcome. Experiment- any process of observation or obtaining data. Examples: tossing dice germination of seed (#of seeds) Experiments have outcomes. Numbers that turn up on dice. whether a particular seed germinated.

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Events Event- is that name we give to each outcome of an experiment that can occur on a single trial. Examples: Toss the dice Numbers 1 through 6 are the complete list of the possible events.

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Events Events are mutually exclusive if any one occurs and its occurrence precludes any other event. Events have observations, elementary units, associated with them. Their sum comprise the population or universe.

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Equiprobable events Equiprobable events- if there is no reason to favor a particular outcome of an experiment, then we should consider all outcomes as equally likely. Toss a fair coin two possible outcomes. Probability of ½ for one side

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This probability is the ratio of the number of ways in which a particular side can turn up divided by the total number of possible outcomes from the toss.

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Apriori Probabilites Aprioir probabilities- ones that were determined by using theory or intuitive judgment. Must have balanced probabilities for equal probable outcomes. Tossing fair die Flipping a fair coin

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Relative Frequency A method for obtaining probabilities when no a priori information is available is called relative frequency. Relative frequency- the number of times a certain event occurs in n trials of an experiment. P(A)= number of events favorable to A number of events in the experiment P(A)= probability of A

21 Basic Properties of Probability 1. The ratio of the number of occurrences of an event A to the total number of trials must fall between 0 and 1 i.e. 0

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Basic Properties of Probability 3. If we examine the nature of A 1,we see that A 1 denotes an event composed of the mutually exclusive events other than A and we call it a compound event. Thus the probability of A 1,P(A 1 ) is the sum of the probabilities of all of the elementary events except A.

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An event with a probability equal to zero means that it is highly unlikely to occur rather than impossible to occur. Likewise, P(A)=1 does not mean that the event is certain to occur, but for all practical purposes it will.

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Relative frequency Relative frequency measures of probability have four basic features: 1. A large number of trials, 2. The relative frequency volume approaches the a priori value if available 3. Use of empirical information gained from experience, and 4. Use of relative frequency to estimate probability.

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Probability in terms of equally likely cases Drawing a random sample 1. Flip a coin 2. Roll a dice Equally likely Rolling a diedie is balanced Flipping a coincoin is fair Dealing cardscards are shuffled thoroughly

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An event is a set of outcomes. Dealing a card which is a spade is an event. Typically an event is a set of outcomes until some interesting property in common. What is the probability of dealing a spade? 13/52

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If there are n equally likely outcomes and an event consists of m outcomes, the probability of the event is m/n. Probability of an ace? 4/52=1/13 Probability of a black card? 26/52=1/2 Probability of a non spade? 39/52=3/4

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Black cards= spades + clubs 26 = 13 + 13 # black cards= # spades + # clubs # cards # cards # cards 26 = 13 + 13 52 52 52 Prob. Black card = Prob. Spade + Prob. Club. No out comes in common.

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Some outcomes in common Event of the card being a spade or a free card. SpadeFCSpade FC 13/5212/523/52 22/52 = 3/52 + 10/52 + 9/52 prob. of prob. of prob. of prob. of card being spade spade face card spade or faced not a not spade faced face card

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Easier to think of outcome in 3 events. No two of which have outcomes in common. An important event is the set of all cards. Probability of 52/52= 1, an event that happens for sure. Probability of a given event + probability of event consisting of all outcomes not in a given event = 1.

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It is important to define the absence of any outcome as the empty event, and its probability is 0/52= 0. It is certain not to happen. Probability of dealing a black card is greater than the probability of a spade.

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Events and Probabilities in General Terms 2 contexts in which the notion of a definite number of equally likely cases does not apply. 1. Where the number of possible outcomes is finite but all outcomes are not equally likely. Coin not fair Spin the needle Whole set of outcomes is not finite Possible states of weather is not finite

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Property 1 0 ≤ Pr (A) ≤ 1 Property 2 Pr (empty event) = 0 Pr (space) = 1

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Addition of Probabilities of Mutually Exclusive Events Two events are mutually exclusive if they have no outcome in common. Spade and Heart being dealt These are mutually exclusive A B A and B are Mutually Exclusive events

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Addition of Probabilities of Mutually Exclusive Events If the events A and B are mutually exclusive, then Pr (A or B) = Pr (A) + Pr (B) Pr (A or B or C) = Pr (A) + Pr (B) + Pr (C)

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Definition The complement of an Event is the event consisting of all outcomes not in that event. 1 = Pr (A) + P (Ā) or P (À) P(Ā) = 1 – Pr (A)

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Addition of Probabilities The event “A and B” Pr (A) = Pr (A and B) + Pr (A and ¯B) Pr (A and B) = Pr (A) – Pr (A and ¯ B) A and B AB

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Terms StatisticsSet Theory Event Set OutcomeMember point element Mutually ExclusiveDisjoint A or BA U B “A union B” A and BA n B “A intersect B” ĀĀ – A complement Empty setnull set

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Relative Frequencies Interpretation of probability: Relation to real life 3 ways Equal probabilities Relative Frequencies Subjective or personal Coin may not be fair Deck may not be shuffled thoroughly

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Relative Frequency More appropriate term in real world. Toss a coin unendingly Pr (head) approaches ½

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Conditional Probabilities The probability of one event given that another event occurs. 100 individuals asked have you seen ad for Bubba burgers? Then asked Did you buy Bubba burgers in the last month?

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Bubba Burger Analysis BuyNot Buy Seen Ad20 (50%)20 (50%) 40 (100%) Not Seen Ad10 (16.7%)50 (83.3%)60 (100%) 30 (30%)70 (70%)100 (100%) BB A A

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Bubba Burger Analysis Draw one person @ random from those who had seen ad, the probability of obtaining a person who bought the bubba burgers is ½ = 20/40 Seen ad 40/100 bought 30/100

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Bubba Burger Analysis Conditional Probability of B given A when Pr (A) > 0 is Pr (B/A) = Pr (A and B) Pr (A) 20/100= 1/2 40/100

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