Assignment 2 (50 points) covers material from lessons 3 - 5

 1. (Lesson 3): Suppose that each of the four randomly selected customers purchasing a hot tub at a certain store chooses either an electric (E) or a gas (G) model.  Assume that these customers make their choices independent of one another and that 40% of all customers select the electric model. P(E) = 0.4

(Show work for the probability questions below)

a.  Find the sample space of hot tub choices for the next four customers.
EEEE
EEEG
EEGE
EEGG
EGEE
EGEG
EGGE
EGGG
GEEE
GEEG
GEGE
GEGG
GGEE
GGEG
GGGE
GGGG
b.  What is the probability that all customers select the electric model?
P(allE) = (0.4)^4
c.  What is the probability that exactly one customer selects the electric model?
P(exactly one E) = 4*(0.4)*(0.6)^3
d.  What is the probability that exactly two customer selects the electric model?
P(exactly two E) = 6*(0.4)^2*(0.6)^2
e.  What is the probability that none of the customers select the electric model?
P(no E) = P(all notE) = (0.6)^4
f.  What is the probability that at least one of the customers selects the electric model?
P(at least one E) = 1 - P(all notE) = 1 - (0.6)^4
g. What is the probability that each customer choses the same model?
P(all E or all G) = (0.4)^4 + (0.6)^4

2. (Lesson 3) Here is a probability table for letter grades in a statistics course. F is failing.

A B C D F
0.20
0.25
0.25
0.20
0.10

If two students are randomly selected from this population...
a. What is the probability that they both failed.
P(all F) = (0.10)^2
b. What is the probability that they both passed?
P(all notF) = (0.90)^2
c. What is the probability that one passed and one failed (i.e. exactly one failed)
2*(0.10)*(0.90)

3. (Lesson 4)
Guy has 0.4 probability of getting into Harvard.
Guy has 0.5 probability of getting into Yale.
Guy has 0.2 probability of getting into both Harvard and Yale.

a. What is the conditional probability that Guy gets into Harvard, if it is known that he got into Yale?
(write the question using the appropriate notation, use a definition, and show the work for your answer)
P(H|Y)=P(HandY)/P(Y)= 0.2/0.5=0.4
b. What is the conditional probability that Guy gets into Harvard, if it is known that he did not get into Yale?
(write the question using the appropriate notation, use a definition, and show the work for your answer)
P(H|notY)=P(H and notY)/P(notY)=[0.4-0.2]/[1-0.5]=0.2/0.5=0.4
c. Are the events "getting into Harvard" and "getting into Yale" independent for Guy? Show mathematically.
Yes they are independent, knowing that Guy gets into Yale has no influence on the probability that he gets into Harvard.
P(H) = P(H|Y)=0.4

4. (Lesson 4) In a class of 25 students, there are 8 females and 10 athletes, and 3 female athletes.  A student is randomly selected from this class. If we select an athlete, what is the probability that the student is a female. (write the question using the appropriate notation, use a definition, and show the work for your answer)
Let F = female student and let A = Athelete
P(F|A)=P(F and A)/P(A) = [3/25]/[10/25]=3/10

5. (Lesson 5) Plant A manufactures 34% of a companies heart pacemakers. Plant B manufactures the rest of this companies heart pacemakers. 0.75% (0.0075) of plant A's pacemakers are defective. Of those from plant B, 0.25% (0.0025) are defective.

a. Tanslate all four sentences above as probability statements.
b. Draw a tree diagram to represent this problem.
c. If one of the pacemakers is randomly selected and it is found to be defective, what is the probability that it was manufactured in plant A? (show the question using notation, show a simple definition, and show the numbers that lead to your answers)

6. (Misc) Under what conditions are the following true:

P(A) + P(B) - P(A and B) = P(A) + P(B)
A and B are mutually exclusive P(A and B) = 0

P(A)P(B|A) = P(A)P(B)
A and B are independent P(B) = P(B|A)

P(B) = [P(A)P(B)]/P(A)
A and B are independent

P(A and B) not = P(A) P(B)
??????????????????????

P(A and notB) = P(A)
A and B are mutually exclusive

7. (Review) Suppose you guess on every question of a 10 question multiple choice test (each question can be A B C or D).

a. What is the probability that you guess the first question correct? (1/4)=0.25
b. What is the probability that you guess all the questions correct? (0.25)^10
c. What is the probability that you guess the first 7 correct, then answer the next 3 wrong? (0.25)^7*(0.75)^3
d. Explain why part c is not the same as the probability of passing (passing is 70%) (you dont have to find a probability here) To pass you need to get at least 7 out of the 10...it does not have to be the first 7 correct...and it does not have to be 7...could be 8 or 9 etc.