# The Law of Total Probability

```Conditional Probability
Conditional Probability
The Law of Total Probability
Let A1 , A2 , . . . , Ak be mutually exclusive and exhaustive events.
Then for any other event B,
P(B) = P(B | A1 ) &middot; P(A1 ) + P(B | A2 ) &middot; P(A2 ) + &middot; &middot; &middot;
+ P(B | Ak ) &middot; P(Ak )
=
k
X
P(B | Ai ) &middot; P(Ai )
i=1
where exhaustive means A1 ∪ A2 ∪ &middot; &middot; &middot; Ak = S.
Conditional Probability
Conditional Probability
Bayes’ Theorem
Let A1 , A2 , . . . , Ak be a collection of k mutually exclusive and
exhaustive events with prior probabilities P(Ai )(i = 1, 2, . . . , k).
Then for any other event B with P(B) &gt; 0, the posterior
probability of Aj given that B has occurred is
P(Aj | B) =
P(B | Aj ) &middot; P(Aj )
P(Aj ∩ B)
= Pk
P(B)
i=1 P(B | Ai ) &middot; P(Ai )
j = 1, 2, . . . k
Independence
Independence
Definition
Two events A and B are independent if P(A | B) = P(A), and
are dependent otherwise.
Independence
Definition
Two events A and B are independent if P(A | B) = P(A), and
are dependent otherwise.
Independence
Independence
The Multiplication Rule for Independent Events
Proposition
Events A and B are independent if and only if
P(A ∩ B) = P(A) &middot; P(B)
Independence
Independence
Independence of More Than Two Events
Definition
Events A1 , A2 , . . . , An are mutually independent if for every k
(k = 2, 3, . . . , n) and every subset of indices i1 , i2 , . . . , ik ,
P(Ai1 ∩ Ai2 ∩ &middot; &middot; &middot; ∩ Aik ) = P(Aii ) &middot; P(Ai2 ) &middot; &middot;&middot;&middot; &middot; P(Aik ).
Random Variables
Random Variables
Definition
For a given sample sample space S of some experiment, a random
variable (rv) is any rule that associates a number with each
outcome in S. In mathematical language, a random variable is a
function whose domain is the sample space and whose range is the
set of real numbers.
Random Variables
Definition
For a given sample sample space S of some experiment, a random
variable (rv) is any rule that associates a number with each
outcome in S. In mathematical language, a random variable is a
function whose domain is the sample space and whose range is the
set of real numbers.
We use uppercase letters, such as X and Y to, denote random
variables and use lowercase letters, such as x and y , to denote
some particular value of the corresponding random variable. For
example, X (s) = x means that value x is associated with the
oucome s by the rv X .
Random Variables
Definition
For a given sample sample space S of some experiment, a random
variable (rv) is any rule that associates a number with each
outcome in S. In mathematical language, a random variable is a
function whose domain is the sample space and whose range is the
set of real numbers.
We use uppercase letters, such as X and Y to, denote random
variables and use lowercase letters, such as x and y , to denote
some particular value of the corresponding random variable. For
example, X (s) = x means that value x is associated with the
oucome s by the rv X .
Random Variables
Random Variables
Examples:
Random Variables
Examples:
1. Assume we toss a coin. Then S = {H, T}. We can define a rv
X by
X (H) = 1 and X (T) = 0
Random Variables
Examples:
1. Assume we toss a coin. Then S = {H, T}. We can define a rv
X by
X (H) = 1 and X (T) = 0
2. A techincian is going to check the quality of 10 prodcuts. For
each product the outcome is either successful (S) or defective (D).
Then we can define a rv Y by
(
1, successful
Y =
0, defective
Random Variables
Examples:
1. Assume we toss a coin. Then S = {H, T}. We can define a rv
X by
X (H) = 1 and X (T) = 0
2. A techincian is going to check the quality of 10 prodcuts. For
each product the outcome is either successful (S) or defective (D).
Then we can define a rv Y by
(
1, successful
Y =
0, defective
Definition
Any random variable whose only possible values are 0 and 1 is
called a Bernoulli random variable.
Random Variables
Random Variables
More examples:
3. (Example 3.3) We are investigating two gas stations. Each has
six gas pumps. Consider the experiment in which the number of
pumps in use at a particular time of day is determined for each of
the stations.
Define rv’s X , Y and U by
X = the total number of pumps in use at the two stations
Y = the difference between the number of pumps in use at station 1
and the number in use at station 2
U = the maximum of the numbers of pumps in use at the two station
Random Variables
More examples:
3. (Example 3.3) We are investigating two gas stations. Each has
six gas pumps. Consider the experiment in which the number of
pumps in use at a particular time of day is determined for each of
the stations.
Define rv’s X , Y and U by
X = the total number of pumps in use at the two stations
Y = the difference between the number of pumps in use at station 1
and the number in use at station 2
U = the maximum of the numbers of pumps in use at the two station
If this experiment is performed and s = (3, 4) results, then
X ((3, 4)) = 3 + 4 = 7, so we say that the observed value of X was
x = 7. Similarly, the observed value of Y would be
y = 3 − 4 = −1, and the observed value of U would be
u = max(3, 4) = 4.
Random Variables
Random Variables
More examples:
4. Assume we toss a coin until we get a Head. Then the sample
space would be S = {H, TH, TTH, TTTH, . . . } If we define a rv
X by X
X = the number we totally tossed
Then X ({H}) = 1, X ({TH}) = 2, X ({TTH}) = 3, . . . , and so on.
Random Variables
More examples:
4. Assume we toss a coin until we get a Head. Then the sample
space would be S = {H, TH, TTH, TTTH, . . . } If we define a rv
X by X
X = the number we totally tossed
Then X ({H}) = 1, X ({TH}) = 2, X ({TTH}) = 3, . . . , and so on.
In this case, the random variable X can be any positive integer,
which in all is infinite.
Random Variables
More examples:
4. Assume we toss a coin until we get a Head. Then the sample
space would be S = {H, TH, TTH, TTTH, . . . } If we define a rv
X by X
X = the number we totally tossed
Then X ({H}) = 1, X ({TH}) = 2, X ({TTH}) = 3, . . . , and so on.
In this case, the random variable X can be any positive integer,
which in all is infinite.
5. Assume we are going to measure the length of 100 desks.
Define the rv Y by
Y = the length of a particular desk
Y can also assume infinitly possible values.
Random Variables
Random Variables
Definition
A dicrete random variable is an rv whose possible values either
constitute a finite set or else can be listed in an infinite sequence in
which there is a first element, a second element, and so on
(“countably” infinite).
Random Variables
Definition
A dicrete random variable is an rv whose possible values either
constitute a finite set or else can be listed in an infinite sequence in
which there is a first element, a second element, and so on
(“countably” infinite).
A random variable is continuous if both of the following apply:
Random Variables
Definition
A dicrete random variable is an rv whose possible values either
constitute a finite set or else can be listed in an infinite sequence in
which there is a first element, a second element, and so on
(“countably” infinite).
A random variable is continuous if both of the following apply:
1. Its set of possible values consists either of all numbers in a
single interval on the number line (possibly infinite in extent, e.g.,
(−∞, ∞) ) or all numbers in a disjoint union of such intervals
(e.g., [0, 10] ∪ [20, 30]).
Random Variables
Definition
A dicrete random variable is an rv whose possible values either
constitute a finite set or else can be listed in an infinite sequence in
which there is a first element, a second element, and so on
(“countably” infinite).
A random variable is continuous if both of the following apply:
1. Its set of possible values consists either of all numbers in a
single interval on the number line (possibly infinite in extent, e.g.,
(−∞, ∞) ) or all numbers in a disjoint union of such intervals
(e.g., [0, 10] ∪ [20, 30]).
2. No possible value of the variable has positive probability, that is,
P(X = c) = 0 for any possible value c.
Examples
Probability Distributions for Discrete RV
Probability Distributions for Discrete RV
An example:
Assume we toss a coin 3 times and record the outcomes. Let Xi be
a random variable defined by
(
1, if the i th outcome is Head;
Xi =
0, if the i th outcome is Tail;
Let X be the random variable such that X = X1 + X2 + X3 , then
X represents the total number of Heads we could get from the
experiment.
Probability Distributions for Discrete RV
An example:
Assume we toss a coin 3 times and record the outcomes. Let Xi be
a random variable defined by
(
1, if the i th outcome is Head;
Xi =
0, if the i th outcome is Tail;
Let X be the random variable such that X = X1 + X2 + X3 , then
X represents the total number of Heads we could get from the
experiment.
If the probability for getting a Head for each toss is 0.7, then the
probabilities for all the outcomes are tabulated as following:
s
x
p(x)
HHH
3
0.343
HHT
2
0.147
HTH
2
0.147
HTT
1
0.063
THH
2
0.147
THT
1
0.063
TTH
1
0.063
TTT
0
0.027
Probability Distributions for Discrete RV
Probability Distributions for Discrete RV
Example continued:
s
HHH HHT
x
3
2
p(x) 0.343 0.147
HTH
2
0.147
HTT
1
0.063
THH
2
0.147
THT
1
0.063
TTH
1
0.063
TTT
0
0.027
Probability Distributions for Discrete RV
Example continued:
s
HHH HHT HTH HTT THH
x
3
2
2
1
2
p(x) 0.343 0.147 0.147 0.063 0.147
We can re-tabulate it only for the x values:
0
1
2
3
x
p(x) 0.027 0.189 0.441 0.343
THT
1
0.063
TTH
1
0.063
TTT
0
0.027
Probability Distributions for Discrete RV
Example continued:
s
HHH HHT HTH HTT THH
x
3
2
2
1
2
p(x) 0.343 0.147 0.147 0.063 0.147
We can re-tabulate it only for the x values:
0
1
2
3
x
p(x) 0.027 0.189 0.441 0.343
Now we can answer various questions.
THT
1
0.063
TTH
1
0.063
TTT
0
0.027
Probability Distributions for Discrete RV
Example continued:
s
HHH HHT HTH HTT THH THT
x
3
2
2
1
2
1
p(x) 0.343 0.147 0.147 0.063 0.147 0.063
We can re-tabulate it only for the x values:
0
1
2
3
x
p(x) 0.027 0.189 0.441 0.343
Now we can answer various questions.
The probability that there are at most 2 Heads is
TTH
1
0.063
P(X ≤ 2) = P(x = 0 or 1 or 2) = p(0) + p(1) + p(2) = 0.657
TTT
0
0.027
Probability Distributions for Discrete RV
Example continued:
s
HHH HHT HTH HTT THH THT
x
3
2
2
1
2
1
p(x) 0.343 0.147 0.147 0.063 0.147 0.063
We can re-tabulate it only for the x values:
0
1
2
3
x
p(x) 0.027 0.189 0.441 0.343
Now we can answer various questions.
The probability that there are at most 2 Heads is
TTH
1
0.063
P(X ≤ 2) = P(x = 0 or 1 or 2) = p(0) + p(1) + p(2) = 0.657
The probability that the number of Heads are is strictly
between 1 and 3 is
P(1 &lt; X &lt; 3) = P(X = 2) = p(2) = 0.441
TTT
0
0.027
Probability Distributions for Discrete RV
Probability Distributions for Discrete RV
Definition
The probability distribution or probability mass function (pmf)
of a discrete rv is defined for every number x by
p(x) = P(X = x) = P(all s ∈ S : X (s) = x).
Probability Distributions for Discrete RV
Definition
The probability distribution or probability mass function (pmf)
of a discrete rv is defined for every number x by
p(x) = P(X = x) = P(all s ∈ S : X (s) = x).
In words, for every possible value x of the random variable, the
pmf specifies the probability of observing that value when the
experiment
is performed. (The conditions p(x) ≥ 0 and
P
all possible x p(x) = 1 are required for any pmf.)
Probability Distributions for Discrete RV
Probability Distributions for Discrete RV
Example 3.8
Six lots of components are ready to be shipped by a certain
supplier. The number of defective components in each lot is as
follows:
Lot
1 2 3 4 5 6
Number of defectives 0 2 0 1 2 0
One of these lots is to be randomly selected for shipment to a
particular customer. Let X be the number of defectives in the
selected lot.
Probability Distributions for Discrete RV
Example 3.8
Six lots of components are ready to be shipped by a certain
supplier. The number of defective components in each lot is as
follows:
Lot
1 2 3 4 5 6
Number of defectives 0 2 0 1 2 0
One of these lots is to be randomly selected for shipment to a
particular customer. Let X be the number of defectives in the
selected lot.
The three possible X values are 0, 1 and 2. The pmf for X is
3
p(0) = P(X = 0) = P(lot 1 or 3 or 6 is selected) = = 0.500
6
1
p(1) = P(X = 1) = P(lot 4 is selected) = = 0.167
6
2
p(2) = P(X = 2) = P(lot 2 or 5 is selected) = = 0.333
6
Probability Distributions for Discrete RV
Probability Distributions for Discrete RV
Example 3.10:
Consider a group of five potential blood donors — a, b, c, d, and e
— of whom only a and b have type O+ blood. Five blood
smaples, one from each individual, will be typed in random order
until an O+ individual is identified. Let the rv Y = the number of
typings necessary to identify an O+ individual. Then what is the
pmf of Y ?
Probability Distributions for Discrete RV
Probability Distributions for Discrete RV
Example:
Consider whether the next customer coming to a certain gas
station buys gasoline or diesel. Let
(
1, if the customer purchases gasoline
X =
0, if the customer purchases diesel
If 30% of all customers in one month purchase diesel, then the pmf
for X is
p(0) = P(X = 0) = P(nextcustomerbuysdiesel) = 0.3
p(1) = P(X = 1) = P(nextcustomerbuysgasoline) = 0.7
p(x) = P(X = x) = 0 for x 6= 0 or 1
Probability Distributions for Discrete RV
Probability Distributions for Discrete RV
Example:
Consider whether the next customer coming to a certain gas
station buys gasoline or diesel. Let
(
1, if the customer purchases gasoline
X =
0, if the customer purchases diesel
If 100α% of all customers in one month purchase diesel, then the
pmf for X is
p(0) = P(X = 0) = P(nextcustomerbuysdiesel) = α
p(1) = P(X = 1) = P(nextcustomerbuysgasoline) = 1 − α
p(x) = P(X = x) = 0 for x 6= 0 or 1
here α is between 0 and 1.
Probability Distributions for Discrete RV
Probability Distributions for Discrete RV
Definition
Suppose p(x) depends on a quantity that can be assigned any one
of a number of possible values, with each different value
determining a different probability distribution. Such a quantity is
called a parameter of the distribution. The collection of all
probability distributions for different values of the parameter is
called a family of probability distribution.
Probability Distributions for Discrete RV
Definition
Suppose p(x) depends on a quantity that can be assigned any one
of a number of possible values, with each different value
determining a different probability distribution. Such a quantity is
called a parameter of the distribution. The collection of all
probability distributions for different values of the parameter is
called a family of probability distribution.
For the previous example, the quantity α is a parameter. Each
different value of α between 0 and 1 determines a different
member of a family of distributions; two such members are


0.3
p(x) = 0.7


0
if x = 0
if x = 1
otherwise


0.25
p(x) = 0.75


0
if x = 0
if x = 1
otherwise
Probability Distributions for Discrete RV
Probability Distributions for Discrete RV
Example:
Assume we are drawing cards from a 100 well-shuffled cards with
replacement. We keep drawing until we get a ♠. Let p = P({♠}),
i.e. there are 100 &middot; p ♠’s. Assume the successive drawings are
independent and define X = the number of drawings. Then
p(1) = P(X = 1) = P({♠}) = p
p(2) = P(X = 2) = P({♠♠}) = (1 − p) &middot; p
p(3) = P(X = 3) = P({♠♠♠}) = (1 − p) &middot; (1 − p) &middot; p
...
Probability Distributions for Discrete RV
Example:
Assume we are drawing cards from a 100 well-shuffled cards with
replacement. We keep drawing until we get a ♠. Let p = P({♠}),
i.e. there are 100 &middot; p ♠’s. Assume the successive drawings are
independent and define X = the number of drawings. Then
p(1) = P(X = 1) = P({♠}) = p
p(2) = P(X = 2) = P({♠♠}) = (1 − p) &middot; p
p(3) = P(X = 3) = P({♠♠♠}) = (1 − p) &middot; (1 − p) &middot; p
...
A general formula would be
(
(1 − p)x−1 &middot; p
p(x) =
0
x = 1, 2, 3, . . .
otherwise
Probability Distributions for Discrete RV
Probability Distributions for Discrete RV
Example:
Assume we are drawing cards from a 100 well-shuffled cards with
replacement. We keep drawing until we get a ♠. Let p = P({♠}),
i.e. there are 100 &middot; p ♠’s. Assume the successive drawings are
independent and define X = the number of drawings.
Probability Distributions for Discrete RV
Example:
Assume we are drawing cards from a 100 well-shuffled cards with
replacement. We keep drawing until we get a ♠. Let p = P({♠}),
i.e. there are 100 &middot; p ♠’s. Assume the successive drawings are
independent and define X = the number of drawings.
If we know that there are 20 ♠’s, i.e. p = 0.2, then what is the
probability for us to draw at most 3 times? More than 2 times?
Probability Distributions for Discrete RV
Example:
Assume we are drawing cards from a 100 well-shuffled cards with
replacement. We keep drawing until we get a ♠. Let p = P({♠}),
i.e. there are 100 &middot; p ♠’s. Assume the successive drawings are
independent and define X = the number of drawings.
If we know that there are 20 ♠’s, i.e. p = 0.2, then what is the
probability for us to draw at most 3 times? More than 2 times?
P(X ≤ 3) = p(1)+p(2)+p(3) = 0.2+0.2&middot;0.8+0.2&middot;(0.8)2 = 0.488
Probability Distributions for Discrete RV
Example:
Assume we are drawing cards from a 100 well-shuffled cards with
replacement. We keep drawing until we get a ♠. Let p = P({♠}),
i.e. there are 100 &middot; p ♠’s. Assume the successive drawings are
independent and define X = the number of drawings.
If we know that there are 20 ♠’s, i.e. p = 0.2, then what is the
probability for us to draw at most 3 times? More than 2 times?
P(X ≤ 3) = p(1)+p(2)+p(3) = 0.2+0.2&middot;0.8+0.2&middot;(0.8)2 = 0.488
P(X &gt; 2) = p(3)+p(4)+p(5)+&middot; &middot; &middot; = 1−p(1)−p(2) = 1−0.2−0.2&middot;0.8 = 0
Probability Distributions for Discrete RV
Probability Distributions for Discrete RV
Definition
The cumulative distribution function (cdf) F (x) of a discrete rv
X with pmf p(x) is defined for every number x by
X
F (x) = P(X ≤ x) =
p(y )
y :y ≤x
For any number x, F(x) is the probability that the observed value
of X will be at most x.
Probability Distributions for Discrete RV
Definition
The cumulative distribution function (cdf) F (x) of a discrete rv
X with pmf p(x) is defined for every number x by
X
F (x) = P(X ≤ x) =
p(y )
y :y ≤x
For any number x, F(x) is the probability that the observed value
of X will be at most x.
F (x) = P(X ≤ x) = P(X is less than or equal to x)
p(x) = P(X = x) = P(X is exactly equal to x)
Probability Distributions for Discrete RV
Probability Distributions for Discrete RV
Example 3.10 (continued):


0





0.4
F (y ) = 0.7



0.9




1
if
if
if
if
if
y &lt;1
1≤y &lt;2
2≤y &lt;3
3≤y &lt;4
y ≥2
Probability Distributions for Discrete RV
Example 3.10 (continued):


0





0.4
F (y ) = 0.7



0.9




1
if
if
if
if
if
y &lt;1
1≤y &lt;2
2≤y &lt;3
3≤y &lt;4
y ≥2
Probability Distributions for Discrete RV
Probability Distributions for Discrete RV
Example:
Assume we are drawing cards from a 100 well-shuffled cards with
replacement. We keep drawing until we get a ♠. Let α = P({♠}),
i.e. there are 100 &middot; α ♠’s. Assume the successive drawings are
independent and define X = the number of drawings. The pmf
would be
(
(1 − α)x−1 &middot; α x = 1, 2, 3, . . .
p(x) =
0
otherwise
Probability Distributions for Discrete RV
Example:
Assume we are drawing cards from a 100 well-shuffled cards with
replacement. We keep drawing until we get a ♠. Let α = P({♠}),
i.e. there are 100 &middot; α ♠’s. Assume the successive drawings are
independent and define X = the number of drawings. The pmf
would be
(
(1 − α)x−1 &middot; α x = 1, 2, 3, . . .
p(x) =
0
otherwise
Then for any positive interger x, we have
F (x) =
X
y ≤x
p(y ) =
x
x−1
X
X
(1 − α)(y −1) &middot; α = α
(1 − α)y
y =1
(
1 − (1 − α)x
=
0
y =0
x ≥1
x &lt;1
Probability Distributions for Discrete RV
Probability Distributions for Discrete RV
Example:
Assume we are drawing cards from a 100 well-shuffled cards with
replacement. We keep drawing until we get a ♠. Let α = P({♠}),
i.e. there are 100 &middot; α ♠’s. Assume the successive drawings are
independent and define X = the number of drawings. The pmf
would be
Probability Distributions for Discrete RV
Probability Distributions for Discrete RV
pmf =⇒ cdf:
F (x) = P(X ≤ x) =
X
y :y ≤x
p(y )
Probability Distributions for Discrete RV
pmf =⇒ cdf:
F (x) = P(X ≤ x) =
X
y :y ≤x
It is also possible cdf =⇒ pmf:
p(y )
Probability Distributions for Discrete RV
pmf =⇒ cdf:
F (x) = P(X ≤ x) =
X
p(y )
y :y ≤x
It is also possible cdf =⇒ pmf:
p(x) = F (x) − F (x−)
where “x−” represents the largest possible X value that is strictly
less than x.
Probability Distributions for Discrete RV
Probability Distributions for Discrete RV
Proposition
For any two numbers a and b with a ≤ b,
P(a ≤ X ≤ b) = F (b) − F (a−)
where “a−” represents the largest possible X value that is strictly
less than a. In particular, if the only possible values are integers
and if a and b are integers, then
P(a ≤ X ≤ b) = P(X = a or a + 1 or . . . or b)
= F (b) − F (a − 1)
Taking a = b yields P(X = a) = F (a) − F (a − 1) in this case.
Probability Distributions for Discrete RV
Probability Distributions for Discrete RV
Example (Problem 23):
A consumer organization that evaluates new automobiles customarily
reports the number of major defects in each car examined. Let X denote
the number of major defects in a randomly selected car of a certain type.
The cdf of X is as follows:


0
x &lt;0




0.06 0 ≤ x &lt; 1



0.19 1 ≤ x &lt; 2



0.39 2 ≤ x &lt; 3
F (x) =

0.67 3 ≤ x &lt; 4




0.92 4 ≤ x &lt; 5




0.97 5 ≤ x &lt; 6




1
x ≤6
Calculate the following probabilities directly from the cdf: (a)p(2),
(b)P(X &gt; 3) and (c)P(2 ≤ X &lt; 5).
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