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.  Tractors in a showroom window Model 50 Model 60 Model 70

 One way to solve is to list and count all possible combinations of the tractors.      

 We can use a tree diagram as well

 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.

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 )

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

 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.

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

 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

 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

 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

 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.

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.

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.

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

 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.

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

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

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<P(A)<1 if A is one of the mutually exclusive and exhaustive events of an experiment. 2. 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(A 1 )=1-P(A). Thus the sum.

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.

 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.

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.

 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

 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

 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

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

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 + 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

 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.

 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.

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

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

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

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)

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)

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

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

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

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

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?

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

Bubba Burger Analysis Draw one 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

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