Chapters 14 and 15 Probability Basics Probability Fundamentals Counting Rules Applied to the Equally Likely Model.

Chapters 14 and 15Probability Basics

Probability FundamentalsCounting Rules Applied to the

Equally Likely Model

Birthday Problem• What is the smallest number of

people you need in a group so that the probability of 2 or more people having the same birthday is greater than 1/2?

• Answer: 23No. of people 23 30 40 60Probability .507.706.891.994

Probability

•Formal study of uncertainty•The engine that drives Statistics

• Primary objective of Chapters 14 and 15:

1. use the rules of probability to calculate appropriate measures of uncertainty.

2. Learn the probability basics so that we can do Statistical Inference

Introduction• Your favorite basketball team has the ball and trails by 2 points with

little time remaining in the game. Should your team attempt a game-tying 2-pointer or go for a buzzer-beating 3-pointer to win the game? (This situation has often been used in Microsoft job interviews).

• After a touchdown should a coach kick the extra point or go for two?

• On 4th down should your favorite football team punt or try for the first down?

• With a man on first base and no one out, should the manager call for a sacrifice bunt?

• If your favorite basketball team has a 3 point lead with little time left on the clock and the other team has the ball, should your team foul?

A phenomenon is random if individual

outcomes are uncertain, but there is

nonetheless a regular distribution of

outcomes in a large number of repetitions.

Randomness and probabilityRandomness ≠ chaos

Coin toss The result of any single coin toss is

random. But the result over many tosses

is predictable, as long as the trials are

independent (i.e., the outcome of a new

coin flip is not influenced by the result of

the previous flip).

The result of any single coin toss is

random. But the result over many tosses

is predictable, as long as the trials are

independent (i.e., the outcome of a new

coin flip is not influenced by the result of

the previous flip).

First series of tossesSecond series

The probability of heads is 0.5 = the proportion of times you get heads in many repeated trials.

The Laws of Probability

1. Relative frequencyevent probability = x/n, where x=# of occurrences of event of interest, n=total # of observations– Coin, die tossing; nuclear power plants?

• Limitationsrepeated observations not practical

Approaches to Probability

Approaches to Probability (cont.)

2. Subjective probabilityindividual assigns prob. based on personal experience, anecdotal evidence, etc.

3. Classical approachevery possible outcome has equal probability (more later)

Basic Definitions

• Experiment: act or process that leads to a single outcome that cannot be predicted with certainty

• Examples:1. Toss a coin2. Draw 1 card from a standard deck of

cards3. Arrival time of flight from Atlanta to

RDU

Basic Definitions (cont.)

• Sample space: all possible outcomes of an experiment. Denoted by S

• Event: any subset of the sample space S;typically denoted A, B, C, etc.Null event: the empty set Certain event: S

Examples1. Toss a coin once

S = {H, T}; A = {H}, B = {T}2. Toss a die once; count dots on upper

faceS = {1, 2, 3, 4, 5, 6}A=even # of dots on upper face={2, 4, 6}B=3 or fewer dots on upper face={1, 2, 3}

3.Select 1 card from adeck of 52 cards.S = {all 52 cards}

Laws of Probability

1)(,0)(.2

event any for ,1)(0 1.

SPP

AAP

Coin Toss Example: S = {Head, Tail}Probability of heads = 0.5Probability of tails = 0.5

3) The complement of any event A is the event that A does not occur, written as A.

The complement rule states that the probability

of an event not occurring is 1 minus the

probability that is does occur.

P(not A) = P(A) = 1 − P(A)

Tail = not Tail = Head

P(Tail ) = 1 − P(Tail) = 0.5

Probability rules (cont’d)

Venn diagram:

Sample space made up of an event

A and its complement A , i.e.,

everything that is not A.

Birthday Problem• What is the smallest number of

people you need in a group so that the probability of 2 or more people having the same birthday is greater than 1/2?

• Answer: 23No. of people 23 30 40 60Probability .507.706.891.994

Example: Birthday Problem

• A={at least 2 people in the group have a common birthday}

• A’ = {no one has common birthday}

502.498.1)'(1)(

498.365

343

365

363

365

364)'(

:23365

363

365

364)'(:3

APAPso

AP

people

APpeople

Unions: , orIntersections: , and

A

A

Mutually Exclusive (Disjoint) Events

• Mutually exclusive ordisjoint events-no outcomesfrom S in common

A and B disjoint: A B=

A and B not disjoint

A

A

Venn Diagrams

Addition Rule for Disjoint Events

4. If A and B are disjoint events, then

P(A or B) = P(A) + P(B)

Laws of Probability (cont.)

General Addition Rule

5. For any two events A and B

P(A or B) = P(A) + P(B) – P(A and B)

20

For any two events A and B

P(A or B) = P(A) + P(B) - P(A and B)P(A or B) = P(A) + P(B) - P(A and B)

A

B

P(A) =6/13

P(B) =5/13

P(A and B) =3/13

A or B

+_

P(A or B) = 8/13

General Addition Rule

Laws of Probability (cont.)

Multiplication Rule

6. For two independent events A and B

P(A and B) = P(A B) = P(A) × P(B)P(A and B) = P(A B) = P(A) × P(B)

Laws of Probability: Summary

• 1. 0 P(A) 1 for any event A• 2. P() = 0, P(S) = 1• 3. P(A’) = 1 – P(A)• 4. If A and B are disjoint events, then

P(A or B) = P(A) + P(B)• 5. For any two events A and B,

P(A or B) = P(A) + P(B) – P(A and B)• 6. for two independent events A and

B,P(A and B) = P(A) × P(B)

M&M candies

Color Brown Red Yellow Green Orange Blue

Probability 0.3 0.2 0.2 0.1 0.1 ?

If you draw an M&M candy at random from a bag, the candy will have one

of six colors. The probability of drawing each color depends on the proportions

manufactured, as described here:

What is the probability that an M&M chosen at random is blue?

What is the probability that a random M&M is any of red, yellow, or orange?

S = {brown, red, yellow, green, orange, blue}

P(S) = P(brown) + P(red) + P(yellow) + P(green) + P(orange) + P(blue) = 1 P(blue) = 1 – [P(brown) + P(red) + P(yellow) + P(green) + P(orange)]

= 1 – [0.3 + 0.2 + 0.2 + 0.1 + 0.1] = 0.1

P(red or yellow or orange) = P(red) + P(yellow) + P(orange)

= 0.2 + 0.2 + 0.1 = 0.5

Example: college students

• L = {student lives on campus}

• M = {student purchases a meal plan}

P(a student either lives or eats on campus)

= P(L or M) = P(L) + P(M) - P(L and M)

=0.56 + 0.62 – 0.42

= 0.76

Suppose 56% of all students live on campus, 62% of all students purchase a campus meal plan and 42% do both.Question: what is the probability that a randomly selected student either lives OR eats on campus.

The Equally Likely Probability Model

Applications and Counting Methods

Assigning Probabilities

If an experiment has N outcomes, then each outcome has probability 1/N of occurring

If an event A1 has n1 outcomes, then

P(A1) = n1/N

DiceYou toss two dice. What is the probability of the outcomes summing to 5?

There are 36 possible outcomes in S, all equally likely (given fair dice).

Thus, the probability of any one of them is 1/36.

P(the roll of two dice sums to 5) =

P(1,4) + P(2,3) + P(3,2) + P(4,1) = 4 / 36 = 0.111

This is S:

{(1,1), (1,2), (1,3), ……etc.}

We Need Efficient Methods for Counting Outcomes

Counting in “Either-Or” Situations• NCAA Basketball Tournament, 68

teams: how many ways can the “bracket” be filled out?

1. How many games?2. 2 choices for each game3. Number of ways to fill out the bracket:

267 = 1.5 × 1020

• Earth pop. about 6 billion; everyone fills out 100 million different brackets

• Chances of getting all games correct is about 1 in 1,000

Counting Example

In the knock-out stages of a soccer tournament, when a game ends in a tie the winner is determined by a penalty-kick shootout. The shootout, which consists of an alternating sequence of penalty kicks, is won by the team with the greatest goal tally after 5 kicks per team.

A coach has selected the 5 players that will take the penalty kicks in a shootout. In how many ways can the coach designate the order in which the 5 players take the penalty kicks?

Solution

There are 5 players to choose to take the first penalty kick, 4 remaining players to take the second penalty kick, 3 players for the third penalty kick, 2 players for the fourth penalty kick, and 1 player for the fifth penalty kick.

The number of possible arrangements is therefore

5 4 3 2 1 = 120

Efficient Methods for Counting Outcomes

Factorial Notation:n!=12 … n

Examples1!=1; 2!=12=2; 3!= 123=6; 4!

=24;5!=120;Special definition: 0!=1

Factorials with calculators and Excel

Calculator: non-graphing: x ! (second function)graphing: bottom p. 9 T I Calculator Commands(math button)

Excel:Insert function: Math and Trig category, FACT function

Permutations

A B C D EHow many ways can we choose 2

letters from the above 5, without replacement, when the order in which we choose the letters is important?

5 4 = 20

Permutations (cont.)

20)!25(

!5:

45!3

!5

)!25(

!52045

25

PNotation

Permutations with calculator and Excel

Calculatornon-graphing: nPr

Graphingp. 9 of T I Calculator Commands(math button)

ExcelInsert function: Statistical, Permut

Combinations

A B C D EHow many ways can we choose 2

letters from the above 5, without replacement, when the order in which we choose the letters is not important?

5 4 = 20 when order importantDivide by 2: (5 4)/2 = 10 ways

Combinations (cont.)

!)!(

!

102

20

21

45

!2!3

!5

!2)!25(

!525

52

rrn

nC

C

rnnr

ST 311 Powerball Lottery

From the numbers 1 through 20,choose 6 different numbers.

Write them on a piece of paper.

Chances of Winning?

760,38!6)!620(

!20

ies?possibilit ofNumber

important.not order t,replacemen

without 20, from numbers 6 Choose

620206

C

Example: Illinois State Lottery

balls) pong pingmillion 16.5 house, ft (1200

months) 10in second 1about (

165,827,25!6!48

!54

importantnot order t;replacemen

withoutnumbers 54 from numbers 6 Choose

2

654 C

North Carolina Powerball Lottery

Prior to Jan. 1, 2009 After Jan. 1, 2009

:

55!3,478,761

5!50!

:

42!42

1!41!

3,478,761*42

146,107,962

5 from 1- 55

1 from 1- 42 (p'ball #)

:

59!5,006,386

5!54!

:

39!39

1!38!

5,006,386*39

195,249,054

5 from 1- 59

1 from 1- 39 (p'ball #)

The Forrest Gump Visualization of Your Lottery Chances

How large is 195,249,054?$1 bill and $100 bill both 6” in length

10,560 bills = 1 mileLet’s start with 195,249,053 $1 bills

and one $100 bill …… and take a long walk, putting

down bills end-to-end as we go

Raleigh to Ft. Lauderdale…

… still plenty of bills remaining, so continue from …

… Ft. Lauderdale to San Diego

… still plenty of bills remaining, so continue from…

… still plenty of bills remaining, so continue from …

… San Diego to Seattle

… still plenty of bills remaining, so continue from …

… Seattle to New York

… still plenty of bills remaining, so …

… New York back to Raleigh

Go around again! Lay a second path of bills

Still have ~ 5,000 bills left!!

Chances of Winning NC Powerball Lottery?

Remember: one of the bills you put down is a $100 bill; all others are $1 bills.

Put on a blindfold and begin walking along the trail of bills.

Your chance of winning the lottery is the same as your chance of selecting the $100 bill if you stop at a random location along the trail and pick up a bill .

End of Chapters 14 and 15 (part 1)