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 50Model 60Model 70

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

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 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)

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

4P4=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.

6P2=6!/(6-2)!=6!/4!=30 (6)(5)(4)(3)(2)(1) 720 30

(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

wxyz wxywxzwyzxyz

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 diceNumbers 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 cointwo 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 dieFlipping 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 Anumber of events in the

experiment P(A)= probability of A

Basic Properties of Probability1. 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(A1)=1-P(A). Thus the sum.

Basic Properties of Probability3. 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.

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 available3. Use of empirical information gained from experience, and4. Use of relative frequency to estimate probability.

Probability in terms of equally likely casesDrawing a random sample

1. Flip a coin2. Roll a dice

Equally likelyRolling a die die is balancedFlipping a coin coin is fairDealing cards cards 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 + clubs26 = 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.

Some outcomes in common Event of the card being a spade or a free card.

Spade FC Spade FC13/52 12/52 3/52

22/52 = 3/52 + 10/52 + 9/52prob. 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 MutuallyExclusive 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 BA B

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” Ā Ā – A complement Empty set null 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

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%)

B B

A

A

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

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