Question 1. Suppose a couple planned to have three children. Let X be the number of girls the couple has.
(a) List all possible arrangements of girls and boys. For example, if the couple had 2 girls and then a boy, then type GGB. (Separate each item in your list with a comma.):
(b) List the sample space for X . (i.e. List the possible values that X may take separated by commas.): .
(c) What is the probability that X = 1?
(d) Find the probability that the couple have three boys:
Question 2.
Two fair dice are rolled 5 times. Let the random variable X represent the number of times that the sum 11 occurs. The table below describes the probability distribution. Find the value of the missing probability.
|
x
|
P(x)
|
|
0
|
0.75141884282545
|
|
1
|
0.221005542007485
|
|
2
|
|
|
3
|
0.00152945011769886
|
|
4
|
4.49838269911429e-05
|
|
5
|
5.29221494013446e-07
|
Would it be unusual to roll a pair of dice 5 times and get no 11s?
Question 3.
The mean and standard deviation of a random variable x are 7 and 3 respectively. Find the mean and standard deviation of the given random variables:
(1) y = x + 9
µ =
σ =
(2) v = 6x
µ =
σ =
(3) w = 6x + 9
µ =
σ =
Question 4. A rock concert producer has scheduled an out- door concert. The producer estimates the attendance will de- pend on the weather according to the following table.
|
Weather
|
Attendance
|
Probability
|
|
wet, cold
|
3000
|
0.2
|
|
wet, warm
|
20000
|
0.2
|
|
dry, cold
|
30000
|
0.1
|
|
dry, warm
|
60000
|
0.5
|
(a) What is the expected attendance? answer:
(b) If tickets cost $ 20 each, the band will cost $ 250000, plus $ 55000 for administration. What is the expected profit?
Question 5.
To examine the effectiveness of its four annual advertising promotions, a mail order company has sent a questionnaire to each of its customers, asking how many of the previous year's promotions prompted orders that would not have otherwise been made. The accompanying table lists the probabilities that were derived from the questionnaire, where X is the random variable representing the number of promotions that prompted orders. If we assume that overall customer behavior next year will be the same as last year, what is the expected number of promotions that each customer will take advantage of next year by ordering goods that otherwise would not be purchased?
A previous analysis of historical records found that the mean value of orders for promotional goods is 26 dollars, with the company earning a gross profit of 20% on each order. Calculate the expected value of the profit contribution next year.
The fixed cost of conducting the four promotions is estimated to be 17000 dollars with a variable cost of 6 dollars per customer for mailing and handling costs. What is the minimum number of customers required by the company in order to cover the cost of promotions? (Round your answer to the next highest integer.)
Question 6.
The owner of a small firm has just purchased a personal com- puter, which she expects will serve her for the next two years. The owner has been told that she "must" buy a surge suppres- sor to provide protection for her new hardware against possible surges or variations in the electrical current, which have the ca- pacity to damage the computer. The amount of damage to the computer depends on the strength of the surge. It has been esti- mated that there is a 3% chance of incurring 450 dollar damage, 5% chance of incurring 250 dollar damage, and 13% chance of incurring 100 dollar damage from a surge within the next two years. An inexpensive suppressor, which would provide protec- tion for only one surge, can be purchased. How much should the owner be willing to pay if she makes decisions on the basis of expected value?
Question 7. X is a random variable having a probability distribution with a mean/expected value of E(X ) = 21.3 and a variance of Var(X ) = 26.
Consider the following random variables.
A = 2X
B = 2X 1
C = 1X + 9
Answer parts (a) through (c).
Part (a) Find the expected value and variance of A.
E(A) = (use two decimals)
Var(A) = (use two decimals)
Part (b) Find the expected value and variance of B.
E(B) = (use two decimals)
Var(B) = (use two decimals)
Part (c) Find the expected value and standard deviation of
C.
E(C) = (use two decimals)
SD(C) = (use two decimals)
Question 8. A customs agent discovers 22 identically looking packets of white power in a traveler's suitcase. Unknown to the customs officer, 16 of these packets contain cocaine, the rest do not contain cocaine.
The customs officer is to randomly pick 4 packets and test each one. A random variable X is to count the number of tested packets that test positive for cocaine.
(a) Finish the probability distribution of X below. Use four decimals in each of your entries.
(b) From the distribution you found in part (a), what can you say about the distribution of X ?
The distribution of X is
• ?
• symmetrical
• skewed to the right
• . skewed to the left
(c) How many of the 4 packets do you expect be to test positive for cocaine? Provide the standard deviation as well.
E(X ) = (use two decimals)
Question 9. A hockey player is to take 3 shots on a certain goalie. The probability he will score a goal on his first shot is 0.4. If he scores on his first shot, the chance he will score on his second shot increases by 0.1; if he misses, the chance that he scores on his second shot decreases by 0.1. This pattern contin- ues to on his third shot: If the player scores on his second shot, the probability he will score on his third shot increases by an- other 0.1; should he not score on his second shot, the probability of scoring on the third shot decreases by another 0.1.
A random variable X counts the number of goals this hockey player scores.
(a) Finish the probability distribution of X below. Use four decimals in each of your entries.
(b) How many goals would you expect this hockey player to score? Enter your answer to four decimals.
(c) Find the standard deviation the random variable X . Enter your answer to two decimals.