(a) Compute the probability that you don't lose any money in the game. (b) Compute the expected net earnings.
(a) Compute the probability that you don't lose any money in the game. (b) Compute the expected net earnings.
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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Please answer correctly. Write by hand please and make sure you write the formula you use. Attached is the formula sheet use this please.
Transcribed Image Text:You pay 5 dollars to play a game. In the game, you roll a die. If the number that comes up is even,
you are paid back that amount in dollars. If the number is odd, you are paid back twice that amount
in dollars.
(a) Compute the probability that you don't lose any money in the game.
(b) Compute the expected net earnings.
![Axioms of Probability
Also Note:
1. P(S)=1
For any two events A and B,
2. For any event E, 0< P(E) < 1
P(A) = P(An B) + P(AnB')
3. For any two mutually exclusive events,
and
P(EUF) = P(E) + P(F)
P(AnB) = P(A|B)P(B).
Events A and B are independent if:
Addition Rule
P(EUF) = P(E) + P(F) – P(ENF)
P(A|B) - P(A)
or
Conditional Probability
P(AN B) = P(A)P(B).
P(B|A) =
P(ANB)
P(A)
Bayes' Theorem:
Total Probability Rule
P(AB)P(B)
P(A|B)P(B) + P(A|B')P(B")
P(A) = P(A|B)P(B) + P(A|B')P(B')
P(B|A) =
Similarly.
Similarly,
P(A) -P(A|E1)P(E1) + P(A|E2)P(E2)+
.+ P(A|E)P(E)
P(B|E,)P(E,)
P(B\E,)P(E,)+ P(B\E,)P(E,) + ·…· + P(B\E,)P(E,
P(E\B) =
Probability Mass and Density Functions
Cumulative Distribution Function
• F(z) = P(X S)
If X is a discrete r.v:
lim,- F(r) = 0
P(X = r) = S(x)
Es(z) = 1 (total probability)
• lim, F(x) -1
• F(x) = Eys/(r) if X is a discrete r.v
• P(a < X < b) =- F(b) – F(a)
Expected Value and Variance
Expected Value of a Function of a RV
• Elh(X)] =E, h(x)f(x) if X is a discrete r.v.
• E'h(X)] = [ h(r)f(z)dr if X is a continu-
• E[X] =E, zf(r) if X is a discrete r.v.
• E[X] = S, ={(z)dz if X is a contimuous r.v.
ous r.v.
• Var(X) - E[X] - E[XP
• E'aX + b] = aE[X] + b
• Var(aX + b) = a³Var(X)
• Var(X) = E|(X - E|X])"
Common Discrete Distributions
•X- Bernoulli(p).
if z = 1;
1-p ifz-0
f(2) =
E[X] = P, Var(X) = p(1 – p).
• X - Geometric(p),
S(2) = (1 – p)*-'p, x € {1,2,.., E[X] = }, Var(X) = .
Geometrie Series: E - for 0 <q<1
%3D
• X- Binomial(n, p).
S(z) = (")(1– p)"-Y,IE {0,1,..., n}, E[X] = np, Var(X) = np(1 – p).
• X- Negative Binomial(r, p).
S(2) = ()(1 – p)*-"P,1€ {r,r+1,..} E[X] = 5, Var(X) = .
• X - Hypergeometric(n, M, N),
, EJx] = n \, Var(X) = =N(1- N).
S(z) =
• X- Poisson(M),
S(z) = , z e {0, 1, ..}, E[X] = At, Var(X)= t.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7b0a5411-bdb0-4804-8919-e2df25fb5bff%2F59b00343-9a3d-452e-a5e2-ff1a64a95c8b%2F43a32or_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) < 1
P(A) = P(An B) + P(AnB')
3. For any two mutually exclusive events,
and
P(EUF) = P(E) + P(F)
P(AnB) = P(A|B)P(B).
Events A and B are independent if:
Addition Rule
P(EUF) = P(E) + P(F) – P(ENF)
P(A|B) - P(A)
or
Conditional Probability
P(AN B) = P(A)P(B).
P(B|A) =
P(ANB)
P(A)
Bayes' Theorem:
Total Probability Rule
P(AB)P(B)
P(A|B)P(B) + P(A|B')P(B")
P(A) = P(A|B)P(B) + P(A|B')P(B')
P(B|A) =
Similarly.
Similarly,
P(A) -P(A|E1)P(E1) + P(A|E2)P(E2)+
.+ P(A|E)P(E)
P(B|E,)P(E,)
P(B\E,)P(E,)+ P(B\E,)P(E,) + ·…· + P(B\E,)P(E,
P(E\B) =
Probability Mass and Density Functions
Cumulative Distribution Function
• F(z) = P(X S)
If X is a discrete r.v:
lim,- F(r) = 0
P(X = r) = S(x)
Es(z) = 1 (total probability)
• lim, F(x) -1
• F(x) = Eys/(r) if X is a discrete r.v
• P(a < X < b) =- F(b) – F(a)
Expected Value and Variance
Expected Value of a Function of a RV
• Elh(X)] =E, h(x)f(x) if X is a discrete r.v.
• E'h(X)] = [ h(r)f(z)dr if X is a continu-
• E[X] =E, zf(r) if X is a discrete r.v.
• E[X] = S, ={(z)dz if X is a contimuous r.v.
ous r.v.
• Var(X) - E[X] - E[XP
• E'aX + b] = aE[X] + b
• Var(aX + b) = a³Var(X)
• Var(X) = E|(X - E|X])"
Common Discrete Distributions
•X- Bernoulli(p).
if z = 1;
1-p ifz-0
f(2) =
E[X] = P, Var(X) = p(1 – p).
• X - Geometric(p),
S(2) = (1 – p)*-'p, x € {1,2,.., E[X] = }, Var(X) = .
Geometrie Series: E - for 0 <q<1
%3D
• X- Binomial(n, p).
S(z) = (")(1– p)"-Y,IE {0,1,..., n}, E[X] = np, Var(X) = np(1 – p).
• X- Negative Binomial(r, p).
S(2) = ()(1 – p)*-"P,1€ {r,r+1,..} E[X] = 5, Var(X) = .
• X - Hypergeometric(n, M, N),
, EJx] = n \, Var(X) = =N(1- N).
S(z) =
• X- Poisson(M),
S(z) = , z e {0, 1, ..}, E[X] = At, Var(X)= t.
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