Probability Basics and Counting Rules for Uncertainty Measures

Chapters 14 and 15Probability Basics Probability Fundamentals Counting 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 morepeople having the same birthday is greater than 1/2? • Answer: 23 No. of people 23 30 40 60 Probability .507 .706 .891 .994

Probability Formal study of uncertainty The engine that drives Statistics Primary objective of Chapters 14 and 15: use the rules of probability to calculate appropriate measures of uncertainty. 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?

Randomness and probability Randomness ≠ chaos A phenomenon is random if individual outcomes are uncertain, but there is nonetheless a regular distribution of outcomes in a large number of repetitions.

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). Coin toss The probability of heads is 0.5 = the proportion of times you get heads in many repeated trials. First series of tosses Second series

The Laws of Probability Approaches to Probability • Relative frequency event probability = x/n, where x=# of occurrences of event of interest, n=total # of observations • Coin, die tossing; nuclear power plants? • Limitations repeated observations not practical

Approaches to Probability (cont.) • Subjective probability individual assigns prob. based on personal experience, anecdotal evidence, etc. • Classical approach every 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 coin 2. Draw 1 card from a standard deck of cards 3. 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 F Certain event: S

Examples 1. Toss a coin once S = {H, T}; A = {H}, B = {T} 2. Toss a die once; count dots on upper face S = {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} • Select 1 card from a deck of 52 cards. S = {all 52 cards}

Laws of Probability

Coin Toss Example: S ={Head, Tail} Probability of heads = 0.5 Probability of tails = 0.5 Probability rules (cont’d) 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  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 morepeople having the same birthday is greater than 1/2? • Answer: 23 No. of people 23 30 40 60 Probability .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}

Unions: , orIntersections: , and AÇB AÈB

Venn Diagrams Mutually Exclusive (Disjoint) Events A and B disjoint: A  B= • Mutually exclusive or disjoint events-no outcomes from S in common AÇB AÈB A and B not disjoint

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)

A or B General Addition Rule For any two events A and B P(A or B) = P(A) + P(B) - P(A and B) P(A) =6/13 A + P(B) =5/13 _ B P(A and B) =3/13 P(A or B) = 8/13

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)

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 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? 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 What is the probability that a random M&M is any of red, yellow, or orange? P(red or yellow or orange) = P(red) + P(yellow) + P(orange) = 0.2 + 0.2 + 0.1 = 0.5

Example: college students 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. • 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

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

Dice You toss two dice. What is the probability of the outcomes summing to 5? This isS: {(1,1), (1,2), (1,3), ……etc.} 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

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? • How many games? • 2 choices for each game • 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!=12 … n • Examples 1!=1; 2!=12=2; 3!= 123=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 E • How 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.)

Permutations with calculator and Excel • Calculator non-graphing: nPr • Graphing p. 9 of T I Calculator Commands (math button) • Excel Insert function: Statistical, Permut

Combinations A B C D E • How 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 important • Divide by 2: (5  4)/2 = 10 ways

Combinations (cont.)

ST 311 Powerball Lottery From the numbers 1 through 20, choose 6 different numbers. Write them on a piece of paper.

Chances of Winning?

Example: Illinois State Lottery

North Carolina Powerball Lottery Prior to Jan. 1, 2009 After Jan. 1, 2009

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 mile • Let’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…

… San Diego to Seattle … still plenty of bills remaining, so continue from …

… Seattle to New York … still plenty of bills remaining, so continue from …

… New York back to Raleigh … still plenty of bills remaining, so …

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 .

Probability Basics and Counting Rules for Uncertainty Measures