Download Conditional Probability

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Week 06 Thurs .
MAT135 Statistics
Probability
Probability is defined to be:
P =
# favorable outcomes
total # of outcomes
Usually expressed as a %
Odds
# unfavorable outcomes
Odds =
# favorable outcomes
Usually expressed as #unfav:#fav
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) = ?
ODDS
IN-CLASS PROBLEM 1
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) = 1/3 or 33%
Horse
Odds
Bodemeister
4
Union Rags
5
Gemologist
8
Take Charge Indy
11
Creative Cause
11
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
IN-CLASS PROBLEMS 2,3
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?
Horse
Odds
Bodemeister
4
Union Rags
5
Gemologist
8
Take Charge Indy
11
Creative Cause
11
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
IN-CLASS PROBLEM 4
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…)
Ex:
Odds
You bet $2 at odds of 20:1
You win!
You will get a profit of:
$2 x 20 = $40
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
Horse
Odds
Bodemeister
4
Union Rags
5
Gemologist
8
Take Charge Indy
11
Creative Cause
11
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
IN-CLASS PROBLEM 5
Kentucky Derby 2012
What would be the (theoretical) payout on a
$1 bet on I’ll Have Another ?
Horse
Odds
Bodemeister
4
Union Rags
5
Gemologist
8
Take Charge Indy
11
Creative Cause
11
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
IN-CLASS PROBLEMS 6,7
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)?
ODDS
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?
What would be the odds for a
specific number?
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
Conditional Probability
But… in the universe we live in,
one event often changed the
likelihood of another even
happening
Conditional Probability
Conditional Probability
Conditional Probability
Notation:
P(A|B)
means:
the probability of A given B
Conditional Probability
The formula:
P(A and B)
P(A|B) =
P(B)
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
www.algebralab.org
slow growth
zero growth
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 and 65)
P(70|65) =
P(65)
P(70)
=
P(65)
Conditional Probability
P(70 and 65)
P(70|65) =
P(65)
P(70)
=
P(65)
A little algebra and:
Conditional Probability
P(70)
P(70|65) × P(65) =
× P(65)
P(65)
So:
P(70|65) × P(65) = P(70)
Conditional Probability
P(70)
P(70|65) × P(65) =
× 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:
Males
Females
Blue
43%
34%
Grey
7%
3%
Green
17%
24%
Brown
33%
39%
CONDITIONAL PROBABILITY
IN-CLASS PROBLEM 16
Males
Females
Blue
43%
34%
Grey
7%
3%
Green
17%
24%
Brown
33%
39%
What is the probability your
subject is a female given your
subject has blue eyes?
CONDITIONAL PROBABILITY
IN-CLASS PROBLEM 16
Males
Females
Blue
22%
17%
Grey
3%
2%
Green
8%
12%
Brown
16%
20%
P(female and blue)
P(female|blue)=
P(blue)
CONDITIONAL PROBABILITY
IN-CLASS PROBLEM 16
Males
Females
Blue
22%
17%
Grey
3%
2%
Green
8%
12%
Brown
16%
20%
P(female and blue)
P(female|blue)=
P(blue)
This time we know
P(female and blue) = 17%
We need P(blue)!
How can we find it?
CONDITIONAL PROBABILITY
IN-CLASS PROBLEM 16
Males
Females
Blue
22%
17%
Grey
3%
2%
Green
8%
12%
Brown
16%
20%
P(female and blue)
P(female|blue)=
P(blue)
P(blue) = 22% + 17% = 39%
CONDITIONAL PROBABILITY
IN-CLASS PROBLEM 16
Males
Females
Blue
22%
17%
Grey
3%
2%
Green
8%
12%
Brown
16%
20%
P(female and blue)
P(female|blue)=
P(blue)
= .17/.39
≈ .436 OR 43.6%
Questions?
You survived!
Turn in your homework!
Don’t forget
your homework
due next week!
Have a great
rest of the week!
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