must show all the work must be completed by 6 pm today!!!

customers a chance to win a small prize for ever bottle of soda they buy. There is a 20%

chance that a customer will find a winning icon at the bottom of the cap upon opening up a

bottle of soda. The customer can then redeem that bottle cap for a small prize. Now, if yours

truly buys a 6-pack of soda, what is the probability that I will win something, i.e., at least

winning a single small prize?

2. (6 points) A department store manager has decided that dress code is necessary for team

coherence. Team members are required to wear either blue shirts or red shirts. There are 9

men and 6 women in the team. On a particular day, 4 men wore blue shirts and 5 other wore

red shirts, whereas 3 women wore blue shirts and 3 others wore red shirt. Apply the Addition

Rule to determine the probability of finding men or blue shirts in the team.

3. (6 points) A consulting company wants to estimate the proportion of Americans who own

their house. What sample size should be obtained if the estimate is expected to be within 0.04

with 95% confidence if

a. they use an estimate of 0.675 from the Census Bureau?

P=0.675

1-p=0.325

E=0.04

Sample size is 527

b. they do not use any prior estimates? But in solving this problem, you are actually using a

form of "prior" estimate in the formula used. In this case, what is your "actual" prior

Estimate? Please explain.

Actual estimate sis 0.675 which is the proportion of American who own their house.

4. (6 points) Most of us love peaches, but hate buying those that are picked too

early. Unfortunately, by waiting until the peaches are almost ripe to pick carries a risk of

having 30% of the picked rot upon arrival at the packing facility. If the packing process is all

done by machines without human inspection to pick out any rotten peaches, what would be

the probability of having at most 4 rotten peaches packed in a box of 12?

5. (6 points) There is a screening test for a rare disease that affects 1.5% of the population.

Unfortunately, the reliability of this screening test is only 70%. What it means is that it gives

a false positive result 30% of the time. Fortunately, there is no false negative. Suppose if

you are tested positive for this rare disease, what is the probability that you are actually

inflicted by this rare disease? (Hint: Bayes’ Theorem)

Percentage of population affected = 1.5%

Reliability = 70%

Percentage of false positive result = 30%

Note that there is no false negative.

Let A1, A2, ... , An be a set of mutually exclusive events that together form the sample space S.

Let B be any event from the same sample space, such that P(B) > 0. Then,

Plug in the given values:

6. (6 points) Assume that you toss a fair six-faced die two times.

(a) (2 pt) How many possible outcomes are in the sample space? Explain your answer.

First time number of outcome =6,

Second time number of outcome =6

Total outcome = 6*6 =36

Possible outcome are in a sample space =36

(b)(2 pt) What is the probability that the product of the two tosses is at most 4? (Show

work and write the answer in simplest fraction form)

Possible outcome that product of the two tosses is at most 4 is 8

(c) (2 pt) What is the probability that the product of the two tosses is at most 4, given that

you get a number less than 3 in the first toss? (Show work and write the answer in

simplest fraction form)

Total Number of outcome that less than 3 in frst outcome =12

Probability given

7. (6 points) We have 7 boys and 4 girls in our school choir. There is an upcoming concert in

the local town hall. Unfortunately, we can only have 6 youths in this performance. This

performance team of 6 has to be picked randomly from the choir of 7 boys and 4 girls.

a. What is the probability that all 4 girls are picked in this team of 6?

 Probability that all 3 girls picked = (4C4)*(7C2)/11C 6= 0.0455

b. What is the probability that none of the girls are picked in this team of 6?

Probability that none of the girls are picked = 1 –P(all girls picked)=  1-0.0455 = 0.9545

c. What is the probability that 2 of the girls are picked in this team of 6?

Probability that 2 of the girls are picked = (4C2)*(7C4)/11C6 = 0.4546

8. (6Pts) A random variable X follows a continuous distribution, which is symmetric to 0. If

P(X ≤-3)= 0.2, then which of the following statements are true? (There are more than one true

statements)

(a) P(X≤-4)≤0.2 True

(b) P(-3≤X≤3)=0.6 True

(c) P(X≤-2)≤0.2

(d) P(X≥3)=0.8

9. (10 points) Below is a summary of the Quiz 1 for two sections of STAT 200 last spring. The

questions and possible maximum scores are different in these two sections. We notice that

Student A4 in Section A and Student B2 in Section B have the same numerical score.

Section A

Student Score

Section B

Student Score

A1 70 B1 15

A2 42 B2 61

A3 53 B3 48

A4 61 B4 90

A5 22 B5 85

A6 87 B6 73

A7 59 B7 48

----- ------ B8 39

How do these two students stand relative to their own classes? And, hence, which student

performed better? Explain your answer.

Both students dud higher than the average.

Section A average is 56

Section B average is 57

Student A4 perform better in relation to their own class

10. (6 points) Mimi plans to make a random guess at 10 true-or-false questions in a test.

Answer the following questions:

(a) Assume random number X is the number of correct answers Mimi gets. As we know, X

follows a binomial distribution. What is n (the number of trials), p (probability of success in

each trial) and q (probability of failure in each trial)?

(b)In order to pass the test, Mimi has to get at least 6 correct answers. What is the

probability that she passes the test? (Show work and round the answer to 4 decimal places)

(c) How many correct answers can she expect to get? (Hint : What is the expected value of

the binomial distribution?) (Show work)

11 (6Pts) Men’s weights are normally distributed with mean 170 pounds and standard deviation

20 pounds.

(a)Find the probability that a randomly selected man has a weight between 145 and 200.

(Show work and round the answer to 4 decimal places)

(b) What is the 70th percentile for men’s weight? (Show work and round the answer to 2

decimal places)

12.(6points) Imagine you are in a game show. There are 30 prizes hidden on a game board with

100 spaces. One prize is worth $100, nine are worth $50, and another twenty are worth $10.

You have to pay $10 to the host if your choice is not correct. Let the random variable x be

the money you get or lose.

(a) (2 pts)Complete the probability distribution

(b) (2pts) What is your expected winning or loss in this game? Be specific in your answer

whether it’s winning or loss.

(c)(2pts) What is the standard deviation of the probability distribution?

13.

(6 points) You need to create a 4-character password for your safe. The first two characters

have to be two different letters of the alphabet, where each letter is used only once, and the

last two characters have to be two numbers from 0 to 9, where the numbers are not

necessarily different. How many different possible passwords are there? (Show work)

There are   ways for selecting first two characters and  ways for selecting last two characters.

Therefore the number of different possible passwords are    

Therefore the number of different possible passwords are 65000.