1. Welcome to .
Week 06 Thurs .
MAT135 Statistics

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

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

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

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

6. 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) = ?

7. 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%

8. 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?

9. 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?

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

11. 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…)

12. 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

13. 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 ?

14. 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)?

15. 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) $ b) $3.93
c) $ d) $2.93

16. How many (numerical) outcomes?

17. What would be the odds for a specific number?

18. 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?

19. 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?

20. Questions?

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

22. Conditional Probability

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

24. Conditional Probability

25. Conditional Probability

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

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

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

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

30. CONDITIONAL PROBABILITY
IN-CLASS PROBLEM 14
Age Structure graphs
rapid growth slow growth zero growth

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

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

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

34. 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%

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

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

37. 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?

38. 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)?

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

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

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

42. 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…)

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

44. 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) = = .84 or 84%

45. 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%

46. Conditional Probability
Sometimes you know one thing, sometimes another…

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

48. 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%

49. 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%

50. 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%

51. 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%

52. 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%

53. Questions?

54. You survived! Turn in your homework! Don’t forget your homework
due next week!
Have a great
rest of the week!