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The Statisticians Objectives 1.Ask the right questions 2.Collect useful data 3.Summarize the data 4.Make decisions and generalizations based on the data 5.Turn the data and decisions into new knowledge

Population uu u u u u u u u Sample u u u u u Inference Sampling Describe Probability The Frame

Probability 1.What is the probability that a flipped coin comes up heads? 2.What is the probability of a randomly selected card being a king? 3.What is the chance of rolling a 3 or 4 on a die?

Random Experiments Outcomes (minimal results) Events (A, B, C…) Sample Space (S)

E1: Flip a coin once – Outcomes: T or H – Events: A={T} B={H} C={H or T} – S = {T,H}

E2: Flip two coins -Outcomes: (H,H) or (H,T) or (T,H) or (T,T). -Events: -A: {One H, one T} = {(H,T), (T,H)} -B: {at least one H} = {(H,T),(T,H), (H,H)} - S = {(T,T); (T,H);(H,T); (H,H) }

E3: Cast two dice Outcomes: (1,1) or (1,2) or … (6,6) Events: A = {(3,4)} B ={The sum is greater than 7} C = … S = {(1,1);(1,2); … ; (6,6)}

Probability: classical definition All outcomes are equally likely P(A) = # outcomes in A Total # outcomes You can think of the classical definition of probability as a proportion

E4. Draw a card A deck of 52 cards: S={all possible draws} # outcomes in S = 52 P(a King) = 4/52 = 1/13 P(a Heart) = 13/52 = ¼ P(king of Hearts)= 1/52

Flip a Coin Three Times Outcomes HHH HHT HTH HTT THH THT TTH TTT 1.P(HHH) = 1/8 = P(Two Heads) = 3.P(At least 2 Heads) = 3/8 = /8 = ½ = 0.5

Roll Two Dice Outcomes: (1,1)(1,2)(1,3)(1,4)(1,5)(1,6) (2,1)(2,2)(2,3)(2,4)(2,5)(2,6) (3,1)(3,2)(3,3)(3,4)(3,5)(3,6) (4,1)(4,2)(4,3)(4,4)(4,5)(4,6) (5,1)(5,2)(5,3)(5,4)(5,5)(5,6) (6,1)(6,2)(6,3)(6,4)(6,5)(6,6) P(Sum =2) = 1/36 = P(Sum=9) = 4/36 = 1/9 = P(Sum=7) = 6/36 = 1/6 = 0.167

Simple Random Sample 1000 people in population, –250 prefer red to green –300 prefer green to red –The rest don’t care Random person –P(prefers green to red) = 300/1000 = 30% –P(don’t care) = 450/100 = 0.45

Probability Properties 0 ≤ P(A) ≤ 1 P(A) = 0 → A is impossible P(S) = 1

Probability: a general definition P(A) = Size of the Event A Size of the Sample Space S

A S Venn diagrams

Unequal Outcomes Assign a probability to each outcome. All probabilities ≥ 0. P(A) = sum P of each outcome in A All probabilities sum to 1. –P(S) = 1 All probabilities ≤ 1.

Choose a Mascot OscarKermitElmoGrover P P(Kermit or Elmo) = = 0.5 P(Oscar or Kermit or Elmo) = = 0.6 P(Grover) = 0.4

OR Rule If A and B can’t both happen: P(A OR B) = P(A)+P(B) A and B are said to beMutually Exclusive

A B S Mutually Exclusive

A Å B =  A and B cannot both happen Examples: A=“Draw a King” B = “Draw a Queen” A=“Roll a 3”B = “Roll an even number”

E2: Flip two coins -A: {One H, one T} -P(A) = P[(H,T) OR (T,H)}] =1/4+1/4= 1/2 -B: {at least one H} -P(B) = P[(H,T) OR (T,H) OR (H,H)} = 3/4

Complements Complement of A  All outcomes not in A AcAc P(A c ) = 1 – P(A) P(Drawing a card other than an Ace) =1 – 1/13 = 12/13

E2: Flip two coins -B: {at least one H} -P(B) = P[(H,T) OR (T,H) OR (H,H)} = 3/4 Or we could use the complement: - B C = {no H} - P(B C ) = P(T,T) = ¼ → P(B) = 1- P(B C ) = 1-1/4=3/4

AND Rule If A and B are Independent P(A AND B) = P(A)P(B) Independent if A occurs, No affect on if B occurs Examples H and then H 6 and then 2

A and B A B A Å B

Successive Events P(Heads and then Tails) = P(Roll ) = P(Roll 1 and then an even number) =

Probability Rules Not ) 1 - Probability OR & Mutually Exclusive ) Add AND & Independence ) Multiply

OR Rule Mutually Exclusive –Events contain no common outcomes –Intersection is empty –They can’t both happen For mutually exclusive events A,B P(A or B) = P(A) + P(B)

Mutually Exclusive Outcomes are mutually Exclusive –Cast a die: only one number can happen –Flip a coin: only one face shows up Mutually Exclusive events are NOT independent –If A and B are mutually exclusive they cannot both happen → P(A AND B) = 0

2. OR Rule Roll 2 dice, P(Sum is 7 or 9) = 1/6 + 1/9 = 5/18 Flip three coins P(1 H or 3 H) = 3/8 + 1/8 = ½ Draw a card P(K or Q) = 1/13 + 1/13 = 2/13 P(Diamond or Heart) = ¼ + ¼ = 1/2 P(K or Diamond) = ????

General OR Rule For any events A, B P(A or B) = P(A) + P(B) – P(A and B)

A OR B A B A Å B S

P(King or Heart) P(King) = 4/52 = 1/13 P(Heart) = 13/52 = ¼ P(King and Heart) = P(King of Hearts) = 1/52 P(King or Heart) = = P(King) + P(Heart) – P(King and Heart) =4/ /52 – 1/52 = 16/52 = 4/13

Example S A B P(A) = 1/3, P(B)= ¼ and P(A and B) =1/6 Compute P(A c or B)

P(A c or B)? Use general OR rule –P(A c or B) = P(A c ) + P(B) – P(A c and B) Note that –P(A c )= 1- P(A) = 1-1/3 = 2/3 –P(B) = ¼ –P(A c and B)? Not independent

S A ACAC

S A A C and B P(A c and B) = P(B) – P(A and B) P(B) = P(A c and B) + P( A and B) Since they are mutually exclusive Then

Finally –P(A c )= 1- P(A) = 1-1/3 = 2/3 –P(B) = ¼ –P(A c and B) = P(B)-P(A and B) = ¼ - 1/6 = 1/12 From which P(A c or B)= 2/3 + ¼ -1/12 = 5/6