A restaurant randomly chooses to discount one of the 20 items on its menu as a daily special. All items are equally likely to be selected and the selection on each day is independent on other days. Suppose that you just tried out their chicken pot pie on discount today. (a) What is the expected number of days you would have to wait until chicken pot pie is on discount again ( (b) What is the probability that chicken pot pie is first discounted again exactly in 5 days? ( (c) If chicken pot pie is not discounted for the next 5 days, what is the probability that it is first discounted again 10 days from now?
A restaurant randomly chooses to discount one of the 20 items on its menu as a daily special. All items are equally likely to be selected and the selection on each day is independent on other days. Suppose that you just tried out their chicken pot pie on discount today. (a) What is the expected number of days you would have to wait until chicken pot pie is on discount again ( (b) What is the probability that chicken pot pie is first discounted again exactly in 5 days? ( (c) If chicken pot pie is not discounted for the next 5 days, what is the probability that it is first discounted again 10 days from now?
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Transcribed Image Text:A restaurant randomly chooses to discount one of the 20 items on its menu as a daily special. All items
are equally likely to be selected and the selection on each day is independent on other days. Suppose
that you just tried out their chicken pot pie on discount today.
(a) What is the expected number of days you would have to wait until chicken pot pie is on discount
again (
(b) What is the probability that chicken pot pie is first discounted again exactly in 5 days? (
(c) If chicken pot pie is not discounted for the next 5 days, what is the probability that it is first
discounted again 10 days from now?
![Axioms of Probability
Also Note:
1. P(S)=1
For any two events A and B,
2. For any event E, 0< P(E)S1
P(A) = P(AN B) + P(An B')
3. For any two mutually exclusive events,
and
P(EUF) = P(E) + P(F)
P(An B) = P(A|B)P(B).
Events A and B are independent if:
Addition Rule
P(EUF) = P(E) + P(F) - P(En F)
P(A|B) = P(A)
or
Conditional Probability
P(An B) = P(A)P(B).
P(B|A) = PANB
%3!
P(A)
Bayes' Theorem:
Total Probability Rule
P(A|B)P(B)
P(A) = P(A|B)P(B) + P(A|B')P(B')
P(B|A) =
P(A|B)P(B) + P(A|B')P(B')
Similarly.
Similarly,
P(A) =P(A|E)P(E) + P(A|E2)P(E2)+
P(B|E1)P(E)
P(Ei|B) =
...+ P(A|E)P(E)
P(B\E,)P(E,)+ P(B\E,)P(E») + ·…· + P(B\E4)P(E
Probability Mass and Density Functions
Cumulative Distribution Function
• F(r) = P(X Sz)
If X is a discrete r.v:
lim,- F(x) = 0
•
P(X = r) = f(r)
• lim,- F(r) = 1
Es(z) = 1 (total probability)
• F(r) = Eus S(z) if X is a discrete r.v.
• P(a <X sb) = F(b) – F(a)
Expected Value and Variance
Expected Value of a Function of a RV
• E[X] = E,rf(x) if X is a discrete r.v.
• E'h(X)) = E, h(z)f(z) if X is a discrete r.v.
• E[X] = [ rf(r)dr if X is a continuous r.v.
• E'h(X)] = * h(r)f(x)dr if X is a continu-
%3D
ous r.v
• Var(X) = E[X2] – E[X}²
• EļaX + b] = aE[X] +6
• Var(X) = E[(x - EX))")
• Var(aX +b) = a?Var(X)
Common Discrete Distributions
•X - Bernoulli(p).
if r = 1;
f(r) =
11-p if z=0
E[X] = p. Var(X) = p(1 – p).
• X- Geometric(p),
S(x) = (1- p)-'p, z€ {1,2,.), E[X] = Var(X) = .
Geometric Series: Eo -. for 0 <q<1
• X- Binomial(n, p),
S(r) = (")(1 – p)"-"p", r€ (0, 1,., n}, EX] = np, Var(X) = np(1 – p).
%3D
• X- Negative Binomial(r, p),
S(2) = ()(1- p)*-,z€ {r,r+1,.) E[X] = 5, Var(X) = ",
• X - Hypergeometric(n, M, N),
fa)=쁘, 티지] = "불, Var(X) 3극"북 (1-부).
• X- Poisson(AM),
S(1) = , z € {0, 1,.}, E[X] = tt, Var(X) = t.
A (M)"](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7b0a5411-bdb0-4804-8919-e2df25fb5bff%2Fe8c0360a-f3fb-49cc-a955-ba58e0f1b93e%2Fmlsocgl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Axioms of Probability
Also Note:
1. P(S)=1
For any two events A and B,
2. For any event E, 0< P(E)S1
P(A) = P(AN B) + P(An B')
3. For any two mutually exclusive events,
and
P(EUF) = P(E) + P(F)
P(An B) = P(A|B)P(B).
Events A and B are independent if:
Addition Rule
P(EUF) = P(E) + P(F) - P(En F)
P(A|B) = P(A)
or
Conditional Probability
P(An B) = P(A)P(B).
P(B|A) = PANB
%3!
P(A)
Bayes' Theorem:
Total Probability Rule
P(A|B)P(B)
P(A) = P(A|B)P(B) + P(A|B')P(B')
P(B|A) =
P(A|B)P(B) + P(A|B')P(B')
Similarly.
Similarly,
P(A) =P(A|E)P(E) + P(A|E2)P(E2)+
P(B|E1)P(E)
P(Ei|B) =
...+ P(A|E)P(E)
P(B\E,)P(E,)+ P(B\E,)P(E») + ·…· + P(B\E4)P(E
Probability Mass and Density Functions
Cumulative Distribution Function
• F(r) = P(X Sz)
If X is a discrete r.v:
lim,- F(x) = 0
•
P(X = r) = f(r)
• lim,- F(r) = 1
Es(z) = 1 (total probability)
• F(r) = Eus S(z) if X is a discrete r.v.
• P(a <X sb) = F(b) – F(a)
Expected Value and Variance
Expected Value of a Function of a RV
• E[X] = E,rf(x) if X is a discrete r.v.
• E'h(X)) = E, h(z)f(z) if X is a discrete r.v.
• E[X] = [ rf(r)dr if X is a continuous r.v.
• E'h(X)] = * h(r)f(x)dr if X is a continu-
%3D
ous r.v
• Var(X) = E[X2] – E[X}²
• EļaX + b] = aE[X] +6
• Var(X) = E[(x - EX))")
• Var(aX +b) = a?Var(X)
Common Discrete Distributions
•X - Bernoulli(p).
if r = 1;
f(r) =
11-p if z=0
E[X] = p. Var(X) = p(1 – p).
• X- Geometric(p),
S(x) = (1- p)-'p, z€ {1,2,.), E[X] = Var(X) = .
Geometric Series: Eo -. for 0 <q<1
• X- Binomial(n, p),
S(r) = (")(1 – p)"-"p", r€ (0, 1,., n}, EX] = np, Var(X) = np(1 – p).
%3D
• X- Negative Binomial(r, p),
S(2) = ()(1- p)*-,z€ {r,r+1,.) E[X] = 5, Var(X) = ",
• X - Hypergeometric(n, M, N),
fa)=쁘, 티지] = "불, Var(X) 3극"북 (1-부).
• X- Poisson(AM),
S(1) = , z € {0, 1,.}, E[X] = tt, Var(X) = t.
A (M)"
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