§ 9.4 Odds, Conditional Probability and
Expected Value
Odds
Where have we seen odds in real life?
Odds
Where have we seen odds in real life?
Definition
The odds in favor of an event or a proposition are the ratio of the
probability that an event will happen to the probability that it will not
happen.
Odds
Where have we seen odds in real life?
Definition
The odds in favor of an event or a proposition are the ratio of the
probability that an event will happen to the probability that it will not
happen.
Example
What is the probability of rolling a 6 on a fair die?
What are the odds in favor of rolling a 6?
What is the probability of getting a 6?
Odds
Where have we seen odds in real life?
Definition
The odds in favor of an event or a proposition are the ratio of the
probability that an event will happen to the probability that it will not
happen.
Example
What is the probability of rolling a 6 on a fair die?
What are the odds in favor of rolling a 6?
What is the probability of getting a 6?
1
6
What is the probability of not getting a 6?
Odds
Where have we seen odds in real life?
Definition
The odds in favor of an event or a proposition are the ratio of the
probability that an event will happen to the probability that it will not
happen.
Example
What is the probability of rolling a 6 on a fair die?
What are the odds in favor of rolling a 6?
What is the probability of getting a 6?
1
6
What is the probability of not getting a 6?
Odds?
5
6
Odds
Where have we seen odds in real life?
Definition
The odds in favor of an event or a proposition are the ratio of the
probability that an event will happen to the probability that it will not
happen.
Example
What is the probability of rolling a 6 on a fair die?
What are the odds in favor of rolling a 6?
What is the probability of getting a 6?
1
6
What is the probability of not getting a 6?
Odds? 1:5
5
6
What Are The Odds?
Example
Suppose you are at the track and you notice that the favorite for one of
the races is running as a 19:1 long shot. What is the probability the
horse wins?
What Are The Odds?
Example
Suppose you are at the track and you notice that the favorite for one of
the races is running as a 19:1 long shot. What is the probability the
horse wins?
1
20
What Are The Odds?
Example
Suppose you are at the track and you notice that the favorite for one of
the races is running as a 19:1 long shot. What is the probability the
horse wins?
1
20
Example
If the odds the Red Sox will win the World Series are 5:2, what is the
probability they win it all?
What Are The Odds?
Example
Suppose you are at the track and you notice that the favorite for one of
the races is running as a 19:1 long shot. What is the probability the
horse wins?
1
20
Example
If the odds the Red Sox will win the World Series are 5:2, what is the
probability they win it all?
5
7
Expected Value
Can anyone tell me what expected value is?
Expected Value
Can anyone tell me what expected value is?
Definition
In probability theory, the expected value of a random variable is the
weighted average of all possible values that this random variable can
take on. The weights used in computing this average correspond to
the probabilities in case of a discrete random variable.
Formula
P
The expected value is calculated by x · P(x) where x is the outcome
and P(x) is the probability of that outcome.
Expected Value
Can anyone tell me what expected value is?
Definition
In probability theory, the expected value of a random variable is the
weighted average of all possible values that this random variable can
take on. The weights used in computing this average correspond to
the probabilities in case of a discrete random variable.
Formula
P
The expected value is calculated by x · P(x) where x is the outcome
and P(x) is the probability of that outcome.
Note: To do expected value for continuous random variables, we need
calculus and integration.
Expected Value
Can anyone tell me what expected value is?
Definition
In probability theory, the expected value of a random variable is the
weighted average of all possible values that this random variable can
take on. The weights used in computing this average correspond to
the probabilities in case of a discrete random variable.
Formula
P
The expected value is calculated by x · P(x) where x is the outcome
and P(x) is the probability of that outcome.
Note: To do expected value for continuous random variables, we need
calculus and integration.
How is this different from averages?
Expected Value
Can anyone tell me what expected value is?
Definition
In probability theory, the expected value of a random variable is the
weighted average of all possible values that this random variable can
take on. The weights used in computing this average correspond to
the probabilities in case of a discrete random variable.
Formula
P
The expected value is calculated by x · P(x) where x is the outcome
and P(x) is the probability of that outcome.
Note: To do expected value for continuous random variables, we need
calculus and integration.
How is this different from averages?
Important: in the long run ...
Investment Returns
Example
Suppose the following table gives the potential return on an
investment per unit invested, including the money put up initially.
Return
-$3000
$0
$4000
Probability
.1
.3
.6
What is the expected return on this investment?
Investment Returns
Example
Suppose the following table gives the potential return on an
investment per unit invested, including the money put up initially.
Return
-$3000
$0
$4000
Probability
.1
.3
.6
What is the expected return on this investment?
Note that we could make $4000 the first time we invest or never make
$4000 on each unit invested.
Investment Returns
Example
Suppose the following table gives the potential return on an
investment per unit invested, including the money put up initially.
Return
-$3000
$0
$4000
Probability
.1
.3
.6
What is the expected return on this investment?
Note that we could make $4000 the first time we invest or never make
$4000 on each unit invested.
In the long term, on average, we expect to make
−3000(.1) + 0(.3) + 4000(.6) = $2100
Would You Play?
Example
Suppose the following game is presented to you. If the cost to play is
factored into the following table of values, would you play?
Winnings ($)
100
10
1
0
-10
Probability
.02
.05
.40
.03
.50
Would You Play?
Example
Suppose the following game is presented to you. If the cost to play is
factored into the following table of values, would you play?
Winnings ($)
100
10
1
0
-10
Probability
.02
.05
.40
.03
.50
To determine if we should play, we find the expected value.
Would You Play?
Example
Suppose the following game is presented to you. If the cost to play is
factored into the following table of values, would you play?
Winnings ($)
100
10
1
0
-10
Probability
.02
.05
.40
.03
.50
To determine if we should play, we find the expected value.
100(.02) + 10(.05) + 1(.4) + 0(.3) − 10(.5) = −2.1
Would You Play?
Example
Suppose the following game is presented to you. If the cost to play is
factored into the following table of values, would you play?
Winnings ($)
100
10
1
0
-10
Probability
.02
.05
.40
.03
.50
To determine if we should play, we find the expected value.
100(.02) + 10(.05) + 1(.4) + 0(.3) − 10(.5) = −2.1
What does this mean in practical terms?
Would You Play?
Example
Suppose the following game is presented to you. If the cost to play is
factored into the following table of values, would you play?
Winnings ($)
100
10
1
0
-10
Probability
.02
.05
.40
.03
.50
To determine if we should play, we find the expected value.
100(.02) + 10(.05) + 1(.4) + 0(.3) − 10(.5) = −2.1
What does this mean in practical terms?
So, if we played this game over and over, we’d expect to lose $2.10
each time we played. The game is unfair.
Would You Play This One?
Example
Suppose a carnival game pays you 2x the roll of a die if you roll even
but you lose 3x if you roll odd. If this a fair game?
Would You Play This One?
Example
Suppose a carnival game pays you 2x the roll of a die if you roll even
but you lose 3x if you roll odd. If this a fair game?
What would the expected value be if it was a fair game?
Would You Play This One?
Example
Suppose a carnival game pays you 2x the roll of a die if you roll even
but you lose 3x if you roll odd. If this a fair game?
What would the expected value be if it was a fair game?
3
1
1
(4 + 8 + 12 − 3 − 9 − 15) = − = −
6
6
2
Since the expected value is not 0, the game is not fair.
Expected Value and Gambling
Example
Suppose you are at the track and the horse you want to bet on is at 5:1
odds. Suppose that the payoff, however, is $1200. what is the
expected payoff if you win?
Expected Value and Gambling
Example
Suppose you are at the track and the horse you want to bet on is at 5:1
odds. Suppose that the payoff, however, is $1200. what is the
expected payoff if you win?
If the odds are 5:1, then the probability your horse wins is
Expected Value and Gambling
Example
Suppose you are at the track and the horse you want to bet on is at 5:1
odds. Suppose that the payoff, however, is $1200. what is the
expected payoff if you win?
If the odds are 5:1, then the probability your horse wins is 16 .
Expected Value and Gambling
Example
Suppose you are at the track and the horse you want to bet on is at 5:1
odds. Suppose that the payoff, however, is $1200. what is the
expected payoff if you win?
If the odds are 5:1, then the probability your horse wins is 16 .
What is the expected value?
Expected Value and Gambling
Example
Suppose you are at the track and the horse you want to bet on is at 5:1
odds. Suppose that the payoff, however, is $1200. what is the
expected payoff if you win?
If the odds are 5:1, then the probability your horse wins is 16 .
What is the expected value?
1
1200 = $200
6
Expected Value and Gambling
Example
Suppose you are playing roulette and you play the same number all
night. If you win, you get a profit of 20 chips, each of which are a $25
chip. What is your expected winnings?
Expected Value and Gambling
Example
Suppose you are playing roulette and you play the same number all
night. If you win, you get a profit of 20 chips, each of which are a $25
chip. What is your expected winnings?
First, what are the winnings?
Expected Value and Gambling
Example
Suppose you are playing roulette and you play the same number all
night. If you win, you get a profit of 20 chips, each of which are a $25
chip. What is your expected winnings?
First, what are the winnings? 20 · 25 = $500.
Expected Value and Gambling
Example
Suppose you are playing roulette and you play the same number all
night. If you win, you get a profit of 20 chips, each of which are a $25
chip. What is your expected winnings?
First, what are the winnings? 20 · 25 = $500.
What is the probability of winning?
Expected Value and Gambling
Example
Suppose you are playing roulette and you play the same number all
night. If you win, you get a profit of 20 chips, each of which are a $25
chip. What is your expected winnings?
First, what are the winnings? 20 · 25 = $500.
1
What is the probability of winning? 38
.
Expected Value and Gambling
Example
Suppose you are playing roulette and you play the same number all
night. If you win, you get a profit of 20 chips, each of which are a $25
chip. What is your expected winnings?
First, what are the winnings? 20 · 25 = $500.
1
What is the probability of winning? 38
.
So, what are the expected winnings?
Expected Value and Gambling
Example
Suppose you are playing roulette and you play the same number all
night. If you win, you get a profit of 20 chips, each of which are a $25
chip. What is your expected winnings?
First, what are the winnings? 20 · 25 = $500.
1
What is the probability of winning? 38
.
So, what are the expected winnings?
1
× 500 ≈ $13.16
38
Conditional Probability
The idea behind conditional probability is that we know some
information that affects the probability of the situation.
Conditional Probability
The idea behind conditional probability is that we know some
information that affects the probability of the situation.
When we have numbers to work with, then we can use the following
formula:
Conditional Probability
The probability of B given that we know A has already occurred is
P(B|A) =
P(B ∩ A)
P(A)
Example
Example
Find the probability of A given B if P(A) = .2, P(B) = .3 and
P(A ∪ B) = .25.
Example
Example
Find the probability of A given B if P(A) = .2, P(B) = .3 and
P(A ∪ B) = .25.
We are not given P(A ∩ B). Can we find it?
Example
Example
Find the probability of A given B if P(A) = .2, P(B) = .3 and
P(A ∪ B) = .25.
We are not given P(A ∩ B). Can we find it?
We can with the Principle of Inclusion and Exclusion.
P(A∪B) = P(A)+P(B)−P(A∩B) ⇒ P(A∩B) = P(A)+P(B)−P(A∪B)
Example
Example
Find the probability of A given B if P(A) = .2, P(B) = .3 and
P(A ∪ B) = .25.
We are not given P(A ∩ B). Can we find it?
We can with the Principle of Inclusion and Exclusion.
P(A∪B) = P(A)+P(B)−P(A∩B) ⇒ P(A∩B) = P(A)+P(B)−P(A∪B)
What is P(A ∩ B)?
Example
Example
Find the probability of A given B if P(A) = .2, P(B) = .3 and
P(A ∪ B) = .25.
We are not given P(A ∩ B). Can we find it?
We can with the Principle of Inclusion and Exclusion.
P(A∪B) = P(A)+P(B)−P(A∩B) ⇒ P(A∩B) = P(A)+P(B)−P(A∪B)
What is P(A ∩ B)? P(A ∩ B) = .2 + .3 − .25 = .25
Example
Example
Find the probability of A given B if P(A) = .2, P(B) = .3 and
P(A ∪ B) = .25.
We are not given P(A ∩ B). Can we find it?
We can with the Principle of Inclusion and Exclusion.
P(A∪B) = P(A)+P(B)−P(A∩B) ⇒ P(A∩B) = P(A)+P(B)−P(A∪B)
What is P(A ∩ B)? P(A ∩ B) = .2 + .3 − .25 = .25
Now, P(B|A) = what?
Example
Example
Find the probability of A given B if P(A) = .2, P(B) = .3 and
P(A ∪ B) = .25.
We are not given P(A ∩ B). Can we find it?
We can with the Principle of Inclusion and Exclusion.
P(A∪B) = P(A)+P(B)−P(A∩B) ⇒ P(A∩B) = P(A)+P(B)−P(A∪B)
What is P(A ∩ B)? P(A ∩ B) = .2 + .3 − .25 = .25
Now, P(B|A) = what?
P(A|B) =
P(A ∩ B)
.25
=
≈ .83
P(B)
.3
Another Example
Example
Suppose a math teacher gave two tests in a class. 25% of the class
passed both exams and 42% of the test passed the first exam. What
percent of the students who passed the first test then passed the
second test?
Another Example
Example
Suppose a math teacher gave two tests in a class. 25% of the class
passed both exams and 42% of the test passed the first exam. What
percent of the students who passed the first test then passed the
second test?
What are we looking for?
P(Second|First) =
Another Example
Example
Suppose a math teacher gave two tests in a class. 25% of the class
passed both exams and 42% of the test passed the first exam. What
percent of the students who passed the first test then passed the
second test?
What are we looking for?
P(Second|First) =
P(First ∩ Second)
P(First)
Another Example
Example
Suppose a math teacher gave two tests in a class. 25% of the class
passed both exams and 42% of the test passed the first exam. What
percent of the students who passed the first test then passed the
second test?
What are we looking for?
P(Second|First) =
So the conditional probability is ...
P(First ∩ Second)
P(First)
Another Example
Example
Suppose a math teacher gave two tests in a class. 25% of the class
passed both exams and 42% of the test passed the first exam. What
percent of the students who passed the first test then passed the
second test?
What are we looking for?
P(Second|First) =
P(First ∩ Second)
P(First)
So the conditional probability is ...
P(Second|First) =
P(First ∩ Second)
.25
=
= .6
P(First)
.42
So, 60% of the class who passed the first test went on to pass the
second one.
Another Example
Example
A jar contains black and white marbles. Two marbles are chosen
without replacement. The probability of selecting a black marble and
then a white marble is 0.34, and the probability of selecting a black
marble on the first draw is 0.47. What is the probability of selecting a
white marble on the second draw, given that the first marble drawn
was black?
Another Example
Example
A jar contains black and white marbles. Two marbles are chosen
without replacement. The probability of selecting a black marble and
then a white marble is 0.34, and the probability of selecting a black
marble on the first draw is 0.47. What is the probability of selecting a
white marble on the second draw, given that the first marble drawn
was black?
What are we looking for?
Another Example
Example
A jar contains black and white marbles. Two marbles are chosen
without replacement. The probability of selecting a black marble and
then a white marble is 0.34, and the probability of selecting a black
marble on the first draw is 0.47. What is the probability of selecting a
white marble on the second draw, given that the first marble drawn
was black?
What are we looking for?
P(White|Black) =
Another Example
Example
A jar contains black and white marbles. Two marbles are chosen
without replacement. The probability of selecting a black marble and
then a white marble is 0.34, and the probability of selecting a black
marble on the first draw is 0.47. What is the probability of selecting a
white marble on the second draw, given that the first marble drawn
was black?
What are we looking for?
P(White|Black) =
So the conditional probability is ...
P(Black ∩ White)
P(Black)
Another Example
Example
A jar contains black and white marbles. Two marbles are chosen
without replacement. The probability of selecting a black marble and
then a white marble is 0.34, and the probability of selecting a black
marble on the first draw is 0.47. What is the probability of selecting a
white marble on the second draw, given that the first marble drawn
was black?
What are we looking for?
P(White|Black) =
P(Black ∩ White)
P(Black)
So the conditional probability is ...
P(White|Black) =
The probability is 72%.
P(Black ∩ White)
.34
=
= .72
P(Black)
.47
Skipping Class
Example
The probability that you skip classes on a Friday is .03. What is the
probability that you skip classes given that it is Friday?
Skipping Class
Example
The probability that you skip classes on a Friday is .03. What is the
probability that you skip classes given that it is Friday?
What do we want?
Skipping Class
Example
The probability that you skip classes on a Friday is .03. What is the
probability that you skip classes given that it is Friday?
What do we want?
P(Skip|Friday) =
P(Skip ∩ Friday)
P(Friday)
Skipping Class
Example
The probability that you skip classes on a Friday is .03. What is the
probability that you skip classes given that it is Friday?
What do we want?
P(Skip|Friday) =
P(Skip ∩ Friday)
P(Friday)
We aren’t given P(Friday). Or are we ... how many school days are
there?
Skipping Class
Example
The probability that you skip classes on a Friday is .03. What is the
probability that you skip classes given that it is Friday?
What do we want?
P(Skip|Friday) =
P(Skip ∩ Friday)
P(Friday)
We aren’t given P(Friday). Or are we ... how many school days are
there?
So ...
Skipping Class
Example
The probability that you skip classes on a Friday is .03. What is the
probability that you skip classes given that it is Friday?
What do we want?
P(Skip|Friday) =
P(Skip ∩ Friday)
P(Friday)
We aren’t given P(Friday). Or are we ... how many school days are
there?
So ...
.03
= .15
.2
So, there is a 15% you skip given that it Friday.
P(Skip|Friday) =
The Wrong Conditional Probability
Example
Suppose P(A) = .4, P(B) = .25 and P(A|B) = .5. What is P(B|A)?
The Wrong Conditional Probability
Example
Suppose P(A) = .4, P(B) = .25 and P(A|B) = .5. What is P(B|A)?
What do wee need to solve this?
The Wrong Conditional Probability
Example
Suppose P(A) = .4, P(B) = .25 and P(A|B) = .5. What is P(B|A)?
What do wee need to solve this?P(A ∩ B)
The Wrong Conditional Probability
Example
Suppose P(A) = .4, P(B) = .25 and P(A|B) = .5. What is P(B|A)?
What do wee need to solve this?P(A ∩ B)
With a little algebra ...
P(A|B) =
P(A ∩ B)
⇒ P(A ∩ B) = .5 × .25 = .125
P(B)
The Wrong Conditional Probability
Example
Suppose P(A) = .4, P(B) = .25 and P(A|B) = .5. What is P(B|A)?
What do wee need to solve this?P(A ∩ B)
With a little algebra ...
P(A|B) =
P(A ∩ B)
⇒ P(A ∩ B) = .5 × .25 = .125
P(B)
So, now we can find what we want.
The Wrong Conditional Probability
Example
Suppose P(A) = .4, P(B) = .25 and P(A|B) = .5. What is P(B|A)?
What do wee need to solve this?P(A ∩ B)
With a little algebra ...
P(A|B) =
P(A ∩ B)
⇒ P(A ∩ B) = .5 × .25 = .125
P(B)
So, now we can find what we want.
P(B|A) =
The Wrong Conditional Probability
Example
Suppose P(A) = .4, P(B) = .25 and P(A|B) = .5. What is P(B|A)?
What do wee need to solve this?P(A ∩ B)
With a little algebra ...
P(A|B) =
P(A ∩ B)
⇒ P(A ∩ B) = .5 × .25 = .125
P(B)
So, now we can find what we want.
P(B|A) =
P(B ∩ A)
.125
=
= .3125
P(A)
.4
So You Think You’re a Pitcher
Example
You have a baseball game today and you want to pitch, but that
depends on who is coaching. If Brian is coaching, the probability you
start is .5, but if Charlie is coaching, the probability you start is .3.
Brian coaches more often - about 60% of the time. What is the
probability you are the starting pitcher today?
So You Think You’re a Pitcher
Example
You have a baseball game today and you want to pitch, but that
depends on who is coaching. If Brian is coaching, the probability you
start is .5, but if Charlie is coaching, the probability you start is .3.
Brian coaches more often - about 60% of the time. What is the
probability you are the starting pitcher today?
Let B be the event Brian is coaching
Let C be the event Charlie is coaching
Let S be the event you are pitching.
So You Think You’re a Pitcher
Example
You have a baseball game today and you want to pitch, but that
depends on who is coaching. If Brian is coaching, the probability you
start is .5, but if Charlie is coaching, the probability you start is .3.
Brian coaches more often - about 60% of the time. What is the
probability you are the starting pitcher today?
Let B be the event Brian is coaching
Let C be the event Charlie is coaching
Let S be the event you are pitching.
We want P(S). What will that equal?
So You Think You’re a Pitcher
Example
You have a baseball game today and you want to pitch, but that
depends on who is coaching. If Brian is coaching, the probability you
start is .5, but if Charlie is coaching, the probability you start is .3.
Brian coaches more often - about 60% of the time. What is the
probability you are the starting pitcher today?
Let B be the event Brian is coaching
Let C be the event Charlie is coaching
Let S be the event you are pitching.
We want P(S). What will that equal?
P(S) = P(S ∩ B) + P(S ∩ C)
Note that we need both since it says nowhere what the probability of
starting is - just what the chances are depending on which coach is
there.
So You Think You’re a Pitcher
P(S ∩ B) =
So You Think You’re a Pitcher
P(S ∩ B) = P(B) × P(S|B) =
So You Think You’re a Pitcher
P(S ∩ B) = P(B) × P(S|B) = .6 × .5 = .3
So You Think You’re a Pitcher
P(S ∩ B) = P(B) × P(S|B) = .6 × .5 = .3
P(S ∩ C) =
So You Think You’re a Pitcher
P(S ∩ B) = P(B) × P(S|B) = .6 × .5 = .3
P(S ∩ C) = P(C) × P(S|C) =
So You Think You’re a Pitcher
P(S ∩ B) = P(B) × P(S|B) = .6 × .5 = .3
P(S ∩ C) = P(C) × P(S|C) = .4 × .3 = .12
So You Think You’re a Pitcher
P(S ∩ B) = P(B) × P(S|B) = .6 × .5 = .3
P(S ∩ C) = P(C) × P(S|C) = .4 × .3 = .12
So, what is P(S)?
So You Think You’re a Pitcher
P(S ∩ B) = P(B) × P(S|B) = .6 × .5 = .3
P(S ∩ C) = P(C) × P(S|C) = .4 × .3 = .12
So, what is P(S)?
P(S) = .3 + .12 = .42
There is a 42% chance you start.