Welcome to . Week 06 Thurs . MAT135 Statistics

Probability is defined to be: # favorable outcomes total # of outcomes Usually expressed as a % P =

# unfavorable outcomes Odds   # unfavorable outcomes # favorable outcomes Usually expressed as #unfav:#fav Odds =

Odds Since total # of outcomes = # favorable outcomes + # unfavorable outcomes You can calculate odds from probabilities (and vice-versa)

ODDS IN-CLASS PROBLEM 1 2:1 means P(win) = ?

2:1 means 2 unfavorable 1 favorable 3 total P(win) = ? ODDS IN-CLASS PROBLEM 1 2:1 means 2 unfavorable 1 favorable 3 total P(win) = ?

2:1 means 2 unfavorable 1 favorable 3 total P(win) = 1/3 or 33% ODDS IN-CLASS PROBLEM 1 2:1 means 2 unfavorable 1 favorable 3 total P(win) = 1/3 or 33%

Kentucky Derby 2012 IN-CLASS PROBLEMS 2,3 Odds = Horse Odds Bodemeister 4 Union Rags 5 Gemologist 8 Take Charge Indy 11 Creative Cause Dullahan 12 Hansen 13 Daddy Nose Best 14 I'll Have Another 15 Alpha 19 Daddy Long Legs 26 El Padrino 29 Went the Day Well 30 Sabercat 37 Done Talking 39 Rousing Sermon 40 Optimizer 42 Trinniberg Bisnath Parboo 44 Liaison 56 Prospective 57 Kentucky Derby 2012 Odds = $bet on another horse/$bet on this horse All are rescaled so that the divisor is 1 What horse is everyone betting on to win? What horse is being bet on by the fewest # of people?

Kentucky Derby 2012 IN-CLASS PROBLEM 4 I’ll Have Another WON!!! Horse Odds Bodemeister 4 Union Rags 5 Gemologist 8 Take Charge Indy 11 Creative Cause Dullahan 12 Hansen 13 Daddy Nose Best 14 I'll Have Another 15 Alpha 19 Daddy Long Legs 26 El Padrino 29 Went the Day Well 30 Sabercat 37 Done Talking 39 Rousing Sermon 40 Optimizer 42 Trinniberg Bisnath Parboo 44 Liaison 56 Prospective 57 Kentucky Derby 2012 I’ll Have Another WON!!! What is the probability of this horse winning?

Odds Odds payouts are strange creatures… Odds = # unfavorable outcomes # favorable outcomes Tells you how much profit you will make on a bet (maybe…)

Odds Ex: You bet $2 at odds of 20:1 You win! You will get a profit of: The bookie will pay you (payout): $40 + $2 = $42 minus his fee (maybe…)

Odds A lot of times the bookie pays you your profit NOT your original bet plus your profit In this case, the bookie will pay you (payout): $40 minus his fee

Kentucky Derby 2012 IN-CLASS PROBLEM 5 Horse Odds Bodemeister 4 Union Rags 5 Gemologist 8 Take Charge Indy 11 Creative Cause Dullahan 12 Hansen 13 Daddy Nose Best 14 I'll Have Another 15 Alpha 19 Daddy Long Legs 26 El Padrino 29 Went the Day Well 30 Sabercat 37 Done Talking 39 Rousing Sermon 40 Optimizer 42 Trinniberg Bisnath Parboo 44 Liaison 56 Prospective 57 Kentucky Derby 2012 What would be the (theoretical) payout on a $1 bet on I’ll Have Another ?

Kentucky Derby 2012 IN-CLASS PROBLEMS 6,7 Horse Odds Bodemeister 4 Union Rags 5 Gemologist 8 Take Charge Indy 11 Creative Cause Dullahan 12 Hansen 13 Daddy Nose Best 14 I'll Have Another 15 Alpha 19 Daddy Long Legs 26 El Padrino 29 Went the Day Well 30 Sabercat 37 Done Talking 39 Rousing Sermon 40 Optimizer 42 Trinniberg Bisnath Parboo 44 Liaison 56 Prospective 57 Kentucky Derby 2012 Payout on a $1 bet on I’ll Have Another = $1 x 15 + $1 = $16 What horse would have had the best payout (if the horse had won)? What horse would have had the lowest payout (if the horse had won)?

You place a $1 to win bet on a horse that has odds listed as "3" IN-CLASS PROBLEM 8 You place a $1 to win bet on a horse that has odds listed as "3" The horse wins. Yay! Assuming a fee of $0.07 per $1 bet, what does the bookie pay you? a) $2.79 b) $3.93   c) $3.79 d) $2.93

How many (numerical) outcomes? http://www.casinoz.club/userfiles/Koleso1.jpg

What would be the odds for a specific number? http://www.casinoz.club/userfiles/Koleso1.jpg

ODDS IN-CLASS PROBLEM 11 If a casino pays 35 to 1 for a winning bet on any single number on a roulette wheel, it means that it is paying using a probability of 1/36. Is this higher than the actual probability or lower?

ODDS IN-CLASS PROBLEMS 12,13 Will the casino pay more for a rare event (low probability) or a common event (high probability)? Is the casino making money or losing money?

Questions?

Conditional Probability Independent events: One event has no effect on the outcome of another event

Conditional Probability http://www.bellevuerarecoins.com/wp-content/uploads/2013/11/bigstock-Coin-Flip-5807921.jpg

Conditional Probability But… in the universe we live in, one event often changed the likelihood of another even happening

Conditional Probability http://thedailycannibal.com/wp-content/uploads/2014/05/cards-620x329.jpg

Conditional Probability https://www.shutterstock.com/image-photo/young-student-overwhelmed-studying-piles-books-116965057?utm_medium=Affiliate&utm_campaign=Idee%20Inc.&utm_source=77643&irgwc=1 http://dlabprep.com/wp-content/uploads/2013/08/Student-stress-study-1.jpg

Conditional Probability Notation: P(A|B) means: the probability of A given B

Conditional Probability The formula: P(A|B) = P(A and B) P(B)

Survival rate and life expectancy in a population CONDITIONAL PROBABILITY IN-CLASS PROBLEM 14 Survival rate and life expectancy in a population

CONDITIONAL PROBABILITY IN-CLASS PROBLEM 14 Let P(N) be the probability of a newborn reaching the age of N years

CONDITIONAL PROBABILITY IN-CLASS PROBLEM 14 Age Structure graphs rapid growth slow growth zero growth www.algebralab.org

CONDITIONAL PROBABILITY IN-CLASS PROBLEM 14 A study has shown that: P(50) = 91.3% P(55) = 88.1% P(65) = 74.6%

CONDITIONAL PROBABILITY IN-CLASS PROBLEM 14 What is the probability of a 50-year-old man reaching the age of 55?

CONDITIONAL PROBABILITY IN-CLASS PROBLEM 14 What is P(55|50)?

CONDITIONAL PROBABILITY IN-CLASS PROBLEM 14 P(55|50) = P(55 and 50)/P(50) = P(55)/P(50) = 88.1/91.3 ≈ .965 or 96.5%

Conditional Probability A probability that a 50 years old will die within 5 years is then a rather comforting: 1 - .965 = .035 or 3.5%

Conditional Probability However, the probability of dying within the next 5 years grows with age

CONDITIONAL PROBABILITY IN-CLASS PROBLEM 15 So if the probability that a man who just turned 65 will die within 5 years is .16, what is the probability for a man to survive till his 70th birthday?

CONDITIONAL PROBABILITY IN-CLASS PROBLEM 15 So if the probability that a man who just turned 65 will die within 5 years is .16, what is the probability for a man to survive till his 70th birthday? P(70)?

Conditional Probability P(70|65) = P(70 and 65) P(65) = P(70) P(65)

Conditional Probability P(70|65) = P(70 and 65) P(65) = P(70) P(65) A little algebra and:

Conditional Probability P(70|65) × P(65) = P(70) P(65) × P(65) So: P(70|65) × P(65) = P(70)

Conditional Probability P(70|65) × P(65) = P(70) P(65) × P(65) So: P(70|65) × P(65) = P(70) (P(70) is the thing we want…)

Conditional Probability Another rule (from algebra): P(A|B) × P(B) = P(A and B)

CONDITIONAL PROBABILITY IN-CLASS PROBLEM 15 If the probability that a man who just turned 65 will die within 5 years is .16, then: P(70|65) = 1 - .16 = .84 or 84%

CONDITIONAL PROBABILITY IN-CLASS PROBLEM 15 What is P(70)? P(70|65) = P(70 and 65) × P(65) = p(70) × P(65) = .84 × .746 ≈ .627 OR 62.7%

Conditional Probability Sometimes you know one thing, sometimes another…

CONDITIONAL PROBABILITY IN-CLASS PROBLEM 16 Eye color of Germans: Blue Grey Green Brown Males 43% 7% 17% 33% Females 34% 3% 24% 39%

CONDITIONAL PROBABILITY IN-CLASS PROBLEM 16 What is the probability your subject is a female given your subject has blue eyes? Blue Grey Green Brown Males 43% 7% 17% 33% Females 34% 3% 24% 39%

CONDITIONAL PROBABILITY IN-CLASS PROBLEM 16 P(female|blue)= P(female and blue) P(blue) Blue Grey Green Brown Males 22% 3% 8% 16% Females 17% 2% 12% 20%

CONDITIONAL PROBABILITY IN-CLASS PROBLEM 16 P(female|blue)= P(female and blue) P(blue) This time we know P(female and blue) = 17% We need P(blue)! How can we find it? Blue Grey Green Brown Males 22% 3% 8% 16% Females 17% 2% 12% 20%

CONDITIONAL PROBABILITY IN-CLASS PROBLEM 16 P(female|blue)= P(female and blue) P(blue) P(blue) = 22% + 17% = 39% Blue Grey Green Brown Males 22% 3% 8% 16% Females 17% 2% 12% 20%

CONDITIONAL PROBABILITY IN-CLASS PROBLEM 16 P(female|blue)= P(female and blue) P(blue) = .17/.39 ≈ .436 OR 43.6% Blue Grey Green Brown Males 22% 3% 8% 16% Females 17% 2% 12% 20%

Questions?

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