Relationship
between Discrete
Distributions and
Nested Problems
Lecture 16
Relationship between Discrete
Distributions and Nested Problems
STAT 225 Introduction to Probability Models
February 23, 2014
Relationship between
Discrete Distributions
Nested Problems
Whitney Huang
Purdue University
16.1
Agenda
1
2
Relationship between Discrete Distributions
Relationship
between Discrete
Distributions and
Nested Problems
Relationship between
Discrete Distributions
Nested Problems
Nested Problems
16.2
Choosing a Distribution
Relationship
between Discrete
Distributions and
Nested Problems
Relationship between
Discrete Distributions
Nested Problems
16.3
Nested Problems
Relationship
between Discrete
Distributions and
Nested Problems
In some of our examples, we will use a distribution to calculate
a probability, then use that probability as the parameter in a
new distribution
Relationship between
Discrete Distributions
Nested Problems
16.4
Example 40
Relationship
between Discrete
Distributions and
Nested Problems
In a jar there are 200,000,000 coins, 5,000,000 of which are
quarters. You select 50 coins from the jar randomly and without
replacement. Let X be the number of quarters in your sample.
1
What is the distribution of X ?
Relationship between
Discrete Distributions
Nested Problems
16.5
Example 40
Relationship
between Discrete
Distributions and
Nested Problems
In a jar there are 200,000,000 coins, 5,000,000 of which are
quarters. You select 50 coins from the jar randomly and without
replacement. Let X be the number of quarters in your sample.
1
What is the distribution of X ?
2
Find the probability that X is 2
Relationship between
Discrete Distributions
Nested Problems
16.5
Example 40
Relationship
between Discrete
Distributions and
Nested Problems
In a jar there are 200,000,000 coins, 5,000,000 of which are
quarters. You select 50 coins from the jar randomly and without
replacement. Let X be the number of quarters in your sample.
1
What is the distribution of X ?
2
Find the probability that X is 2
3
Is there an approximate distribution for X , why or why not?
Relationship between
Discrete Distributions
Nested Problems
16.5
Example 40 cont’d
Relationship
between Discrete
Distributions and
Nested Problems
Solution.
1
X ∼ Hyp(N = 200, 000, 000, n = 50, K = 5, 000, 000)
Relationship between
Discrete Distributions
Nested Problems
16.6
Example 40 cont’d
Relationship
between Discrete
Distributions and
Nested Problems
Solution.
1
2
X ∼ Hyp(N = 200, 000, 000, n = 50, K = 5, 000, 000)
(5000000
)(195000000
)
2
48
P(X = 2) =
= 0.2271
200000000
( 50 )
Relationship between
Discrete Distributions
Nested Problems
16.6
Example 40 cont’d
Relationship
between Discrete
Distributions and
Nested Problems
Solution.
1
2
3
X ∼ Hyp(N = 200, 000, 000, n = 50, K = 5, 000, 000)
(5000000
)(195000000
)
2
48
P(X = 2) =
= 0.2271
200000000
( 50 )
Yes, because N 20n. The approximation is
X ∗ ∼ Binomioal(n = 50, p = .025)
Relationship between
Discrete Distributions
Nested Problems
16.6
Example 41
Nick plays a game with his friend Eric. Eric bets $1 every hand
(5 cards). If he gets a full house, he wins $500 (on top of
keeping his bet of $1); otherwise, he loses the $1 to Nick.
Suppose in an afternoon of gambling, Nick and Eric play this
game 500 times. Let E denote the number of hands that Eric
wins in this particular afternoon.
1
Name the distribution and parameter(s) for E
Relationship
between Discrete
Distributions and
Nested Problems
Relationship between
Discrete Distributions
Nested Problems
16.7
Example 41
Nick plays a game with his friend Eric. Eric bets $1 every hand
(5 cards). If he gets a full house, he wins $500 (on top of
keeping his bet of $1); otherwise, he loses the $1 to Nick.
Suppose in an afternoon of gambling, Nick and Eric play this
game 500 times. Let E denote the number of hands that Eric
wins in this particular afternoon.
1
Name the distribution and parameter(s) for E
2
Find the probability that E is at least 3
Relationship
between Discrete
Distributions and
Nested Problems
Relationship between
Discrete Distributions
Nested Problems
16.7
Example 41
Nick plays a game with his friend Eric. Eric bets $1 every hand
(5 cards). If he gets a full house, he wins $500 (on top of
keeping his bet of $1); otherwise, he loses the $1 to Nick.
Suppose in an afternoon of gambling, Nick and Eric play this
game 500 times. Let E denote the number of hands that Eric
wins in this particular afternoon.
1
Name the distribution and parameter(s) for E
2
Find the probability that E is at least 3
3
Is an approximation appropriate for E? Why or why not?
Relationship
between Discrete
Distributions and
Nested Problems
Relationship between
Discrete Distributions
Nested Problems
16.7
Example 41
Nick plays a game with his friend Eric. Eric bets $1 every hand
(5 cards). If he gets a full house, he wins $500 (on top of
keeping his bet of $1); otherwise, he loses the $1 to Nick.
Suppose in an afternoon of gambling, Nick and Eric play this
game 500 times. Let E denote the number of hands that Eric
wins in this particular afternoon.
1
Name the distribution and parameter(s) for E
2
Find the probability that E is at least 3
3
Is an approximation appropriate for E? Why or why not?
4
If an approximation is appropriate, find P(E ∗ ≥ 3)
Relationship
between Discrete
Distributions and
Nested Problems
Relationship between
Discrete Distributions
Nested Problems
16.7
Example 41 cont’d
Relationship
between Discrete
Distributions and
Nested Problems
Solution.
1
E ∼ Binomial(n = 500, p) where p = P(full house) =
(131)(43)(121)(42)
= 0.0014
(525)
Relationship between
Discrete Distributions
Nested Problems
16.8
Example 41 cont’d
Relationship
between Discrete
Distributions and
Nested Problems
Solution.
1
2
E ∼ Binomial(n = 500, p) where p = P(full house) =
(131)(43)(121)(42)
= 0.0014
(525)
P(E is at least 3) = P(E ≥ 3) = 1 − P(E ≤ 2) =
1 − (P(E = 0) + P(E = 1) + P(E = 2)) =
1 − (0.4864 + 0.3508 + 0.1263) = 0.0365
Relationship between
Discrete Distributions
Nested Problems
16.8
Example 41 cont’d
Relationship
between Discrete
Distributions and
Nested Problems
Solution.
1
E ∼ Binomial(n = 500, p) where p = P(full house) =
(131)(43)(121)(42)
= 0.0014
(525)
2
P(E is at least 3) = P(E ≥ 3) = 1 − P(E ≤ 2) =
1 − (P(E = 0) + P(E = 1) + P(E = 2)) =
1 − (0.4864 + 0.3508 + 0.1263) = 0.0365
3
An approximation is appropriate since n > 100 and
p < .01. The approximation is
E ∗ ∼ Poisson(λ = np = 0.7203)
Relationship between
Discrete Distributions
Nested Problems
16.8
Example 41 cont’d
Relationship
between Discrete
Distributions and
Nested Problems
Solution.
1
E ∼ Binomial(n = 500, p) where p = P(full house) =
(131)(43)(121)(42)
= 0.0014
(525)
2
P(E is at least 3) = P(E ≥ 3) = 1 − P(E ≤ 2) =
1 − (P(E = 0) + P(E = 1) + P(E = 2)) =
1 − (0.4864 + 0.3508 + 0.1263) = 0.0365
3
An approximation is appropriate since n > 100 and
p < .01. The approximation is
E ∗ ∼ Poisson(λ = np = 0.7203)
4
Relationship between
Discrete Distributions
Nested Problems
P(E ∗ ≥ 3) = 1 − P(E ∗ ≤ 2) =
1 − (P(E ∗ = 0) + P(E ∗ = 1) + P(E ∗ = 2)) =
1 − (0.4866 + 0.3505 + 0.1262) = 0.0367
16.8
Example 42
The wonderful candy shop, Albanese Candy Outlet, makes
chocolate chip cookies as part of their production line.
Chocolate chips in the cookies are randomly and independently
distributed with an average of 12 chocolate chips per cookie.
You and 9 of your friends decide to make a trip to Albanese
Candy Outlet. Each of you buys one chocolate chip cookie.
1
What is the probability that your cookie contains between
10 and 15 chocolate chips inclusive?
Relationship
between Discrete
Distributions and
Nested Problems
Relationship between
Discrete Distributions
Nested Problems
16.9
Example 42
The wonderful candy shop, Albanese Candy Outlet, makes
chocolate chip cookies as part of their production line.
Chocolate chips in the cookies are randomly and independently
distributed with an average of 12 chocolate chips per cookie.
You and 9 of your friends decide to make a trip to Albanese
Candy Outlet. Each of you buys one chocolate chip cookie.
1
What is the probability that your cookie contains between
10 and 15 chocolate chips inclusive?
2
What is the probability that 5 or 6 people in your group
have cookies with between 10 and 15 chocolate chips
inclusive?
Relationship
between Discrete
Distributions and
Nested Problems
Relationship between
Discrete Distributions
Nested Problems
16.9
Example 42
The wonderful candy shop, Albanese Candy Outlet, makes
chocolate chip cookies as part of their production line.
Chocolate chips in the cookies are randomly and independently
distributed with an average of 12 chocolate chips per cookie.
You and 9 of your friends decide to make a trip to Albanese
Candy Outlet. Each of you buys one chocolate chip cookie.
1
What is the probability that your cookie contains between
10 and 15 chocolate chips inclusive?
2
What is the probability that 5 or 6 people in your group
have cookies with between 10 and 15 chocolate chips
inclusive?
3
While examining your cookies (one-by-one), what is the
probability that it takes at least 4 cookies to find the first
one with between 10 and 15 chocolate chips inclusive?
Relationship
between Discrete
Distributions and
Nested Problems
Relationship between
Discrete Distributions
Nested Problems
16.9
Example 42
The wonderful candy shop, Albanese Candy Outlet, makes
chocolate chip cookies as part of their production line.
Chocolate chips in the cookies are randomly and independently
distributed with an average of 12 chocolate chips per cookie.
You and 9 of your friends decide to make a trip to Albanese
Candy Outlet. Each of you buys one chocolate chip cookie.
1
What is the probability that your cookie contains between
10 and 15 chocolate chips inclusive?
2
What is the probability that 5 or 6 people in your group
have cookies with between 10 and 15 chocolate chips
inclusive?
3
While examining your cookies (one-by-one), what is the
probability that it takes at least 4 cookies to find the first
one with between 10 and 15 chocolate chips inclusive?
4
Suppose you and your 9 friends were to go repeatedly to
Albanese Candy Outlet. What is the probability that it
takes until your sixth trip so that 5 or 6 people in your
group have 12 or 13 chocolate chips in their cookie?
Relationship
between Discrete
Distributions and
Nested Problems
Relationship between
Discrete Distributions
Nested Problems
16.9
Example 42 cont’d
Relationship
between Discrete
Distributions and
Nested Problems
Solution.
1
Let X be the number of chips in your cookie then
X ∼ Poisson(λ = 12)
P15
P15
−12
x
P(10 ≤ X ≤ 15) = x=10 P(X = x) = x=10 e x!12 =
0.6020
Relationship between
Discrete Distributions
Nested Problems
16.10
Example 42 cont’d
Relationship
between Discrete
Distributions and
Nested Problems
Solution.
1
2
Let X be the number of chips in your cookie then
X ∼ Poisson(λ = 12)
P15
P15
−12
x
P(10 ≤ X ≤ 15) = x=10 P(X = x) = x=10 e x!12 =
0.6020
Relationship between
Discrete Distributions
Nested Problems
Let Y be the number people in your group have cookies
with between 10 and 15 chocolate chips inclusive then
Y ∼ Binomial(n = 10, p = P(10 ≤ X ≤ 15) = .6020)
P(Y = 5 or 6) = P(Y = 5) + P(Y = 6) =
0.1990 + 0.2508 = 0.4498
16.10
Example 42 cont’d
Relationship
between Discrete
Distributions and
Nested Problems
Solution.
3. Let Z be the number of cookies it takes to find the first one
with between 10 and 15 chocolate chips inclusive then
Z ∼ Geo(p = 0.6020)
P(Z > 3) = (1 − .6020)3 = .0630
4. P(X = 12 or 13) = P(X = 12) + P(X = 13) = 0.2199
Let W ∼ Binomial(n = 10, p = 0.2199) then
P(W = 5 or 6) = P(W = 5) + P(W = 6) = 0.0463
Let Q ∼ Geo(p = 0.0463) then P(Q = 6) = .0365
Relationship between
Discrete Distributions
Nested Problems
16.11
Example 43
Relationship
between Discrete
Distributions and
Nested Problems
An urn contains 6 red balls, 6 green balls, and 3 purple balls.
You randomly reach in and pull out 4 balls.
1
Assume sampling is done with replacement. What is the
probability that you draw at least 2 purple balls?
Relationship between
Discrete Distributions
Nested Problems
16.12
Example 43
Relationship
between Discrete
Distributions and
Nested Problems
An urn contains 6 red balls, 6 green balls, and 3 purple balls.
You randomly reach in and pull out 4 balls.
1
Assume sampling is done with replacement. What is the
probability that you draw at least 2 purple balls?
2
Assume sampling is done without replacement. What is
the probability that you draw at least 2 purple balls?
Relationship between
Discrete Distributions
Nested Problems
16.12
Example 43
Relationship
between Discrete
Distributions and
Nested Problems
An urn contains 6 red balls, 6 green balls, and 3 purple balls.
You randomly reach in and pull out 4 balls.
1
Assume sampling is done with replacement. What is the
probability that you draw at least 2 purple balls?
2
Assume sampling is done without replacement. What is
the probability that you draw at least 2 purple balls?
3
Assume sampling is done with replacement. What is the
probability that it takes you until your tenth sample to get a
sample with at least 2 purple balls?
Relationship between
Discrete Distributions
Nested Problems
16.12
Example 43 cont’d
Relationship
between Discrete
Distributions and
Nested Problems
Solution.
1
X ∼ Binomial(n = 4, p = .2)
P(X ≥ 2) = 1 − (P(X = 0) + P(X = 1)) =
1 − 0.4096 − 0.4092 = 0.1808
Relationship between
Discrete Distributions
Nested Problems
16.13
Example 43 cont’d
Relationship
between Discrete
Distributions and
Nested Problems
Solution.
1
2
X ∼ Binomial(n = 4, p = .2)
P(X ≥ 2) = 1 − (P(X = 0) + P(X = 1)) =
1 − 0.4096 − 0.4092 = 0.1808
Y ∼ Hyp(N = 15, n = 4, K = 3)
P(Y ≥ 2) = 1 − (P(Y = 0) + P(Y = 1)) =
1 − 0.3626 − 0.4835 = 0.1539
Relationship between
Discrete Distributions
Nested Problems
16.13
Example 43 cont’d
Relationship
between Discrete
Distributions and
Nested Problems
Solution.
1
X ∼ Binomial(n = 4, p = .2)
P(X ≥ 2) = 1 − (P(X = 0) + P(X = 1)) =
1 − 0.4096 − 0.4092 = 0.1808
2
Y ∼ Hyp(N = 15, n = 4, K = 3)
P(Y ≥ 2) = 1 − (P(Y = 0) + P(Y = 1)) =
1 − 0.3626 − 0.4835 = 0.1539
3
Z ∼ Geo(p = 0.1808)
P(Z = 10) = (0.1808)(1 − 0.1808)9 = 0.0300
Relationship between
Discrete Distributions
Nested Problems
16.13