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Probability

Probability. Chapter 3. Methods of Counting. The type of counting important for probability theory involves choosing the number of ways we can arrange a set of items. Permutation. Permutation- any ordered sequence of a group or set of things.

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Probability

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  1. Probability Chapter 3

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

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

  4. One way to solve is to list and count all possible combinations of the tractors. • 1. 50 60 70 • 2. 50 70 60 • 3. 60 50 70 • 4. 60 70 50 • 5. 70 50 60 • 6. 70 60 50

  5. We can use a tree diagram as well

  6. 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.

  7. General Rule • If we can perform the first step in N1 ways the second step in N2 ways and so on for r steps, then the total number of ways we can perform the r steps is given by their product. • (n1)(n2)(n3)…(nr)

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

  9. 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 • 4P4=4! (4)(3)(2)(1)= 24 possible comb.

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

  11. 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. • 6P2=6!/(6-2)!=6!/4!=30 • (6)(5)(4)(3)(2)(1) 720 30 (4)(3)(2)(1) 24

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

  13. 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

  14. 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.

  15. 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.

  16. 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.

  17. 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

  18. 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.

  19. 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

  20. 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 • 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<P(A)<1 if A is one of the mutually exclusive and exhaustive events of an experiment. • Since the events are collectively exhaustive, one of the elementary events must occur on a given trial. The probability that an event that occurs in not A is P(A1)=1-P(A). Thus the sum.

  22. Basic Properties of Probability • If we examine the nature of A1,we see that A1 denotes an event composed of the mutually exclusive events other than A and we call it a compound event. Thus the probability of A1,P(A1) is the sum of the probabilities of all of the elementary events except A.

  23. 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.

  24. 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.

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

  26. 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

  27. 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

  28. 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.

  29. Some outcomes in common • Event of the card being a spade or a free card. • Spade FC Spade FC 13/52 12/52 3/52 • 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

  30. 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.

  31. 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.

  32. 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

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

  34. 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 and B are Mutually Exclusive events A B

  35. 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)

  36. 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)

  37. 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 A B

  38. Terms • Statistics Set Theory • Event Set • Outcome Member point element • Mutually Exclusive Disjoint • A or B A U B “A union B” • A and B A n B “A intersect B” • ĀĀ – A complement • Empty set null set

  39. 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

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

  41. 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?

  42. Bubba Burger Analysis B B Buy Not Buy Seen Ad 20 (50%) 20 (50%) 40 (100%) Not Seen Ad 10 (16.7%) 50 (83.3%) 60 (100%) 30 (30%) 70 (70%) 100 (100%) A A

  43. 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

  44. 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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