Download Name Due: Monday, March 27 Homework 16 1. In roulette a player

Name
Due: Monday, March 27
Homework 16
1. In roulette a player may place a $1 bet on any one of the 38 numbers on the roulette wheel. He
wins $35 (plus the return of his bet) if the ball lands on his number; otherwise he loses his bet.
(a) Write down the probability distribution for earnings from this type of bet.
(b) Find the expected value.
2. (a) Last weekend, a movie theater sold 100 adult tickets at $12 each, and 20 childrens tickets at
$4 each. What was their mean profit per ticket?
(b) Last weekend, a movie theater sold x adult tickets at $12 each, and y childrens tickets at $4
each. What was their mean profit per ticket?
3. Three marbles are selected at random from a cup which has six blue marbles and four red marbles.
(a) What is the expected number of blue marbles?
(b) What is the expected number of red marbles?
4. If the data set {5, 6, x} has the same mean as {2, 7, 9} what is x?
5. Student A received the following course grades during her first year at college:
4, 4, 4, 4, 3, 3, 2, 2, 2, 0
Student B received the following course grades during her first year at college:
4, 4, 4, 4, 4, 4, 3, 1, 1, 1
(a) Compute the population means.
(b) Compute the population variances.
(c) Which student had the better grade point average. Explain.
(d) Which student was more consistent. Explain.
6. Twenty-five percent of high schoolers play a musical instrument. Of those who play an instrument,
50% were accepted to private colleges. Of those who do not play a musical instrument, 40% were
accepted to private colleges.
(a) Draw a tree diagram to represent this situation.
(b) What is the probability that a randomly selected student plays an instrument and was accepted to a private college?
(c) What is the probability that a randomly selected student was not accepted to a private
college?
(d) Given that a student plays basketball, what is the probability someone plays an instrument
given that they were not accepted to a private college?
7. Let P r(A) =
1
,
10
P r(B) = 12 , and P r(A ∩ B) =
1
.
20
Find the following:
(a) P r(A ∪ B) =
(b) P r(A|B) =
(c) Are A and B independent events? Explain your answer.
(d) P r(A0 ) =
(e) P r(A ∪ A0 ) =
8. (a) Durring a typical South Bend winter, it will be windy half of the days. If it is windy, my
neighbor Dave complain about the weather 80% of the time. What is the probability that it
will be windy and Dave will complain about the weather?
(b) Johnnie and Bennie are best friends. Both are trying out for the community theater production of Hamlet. Johnnie has a 40% chance of being cast, and Bennie has a 20% chance of
being cast. Assuming that their chances of being cast are independent, what is the probability
that Bennie will be cast and Johnnie will not be cast?
(c) The National Park Service reports that 15% of people are avid hikers. 1% of people are avid
hikers and have visited Zion National Park. Given that a person is an avid hiker, what is the
probability that they have been to Zion National Park?
9. On a typical weekend, the likelihood that I will bake a dessert is .8. Let X be the binomial random
variable associated to the number of weekends I bake a dessert in a typical year (that is, out of
52 weekends).
(a) What is the probability that I bake exactly 50 weekends? That is, what is P r(X = 50)?
(b) What is the expected number of weekends I will bake in a year? That is, the expected value
of the binomial random variable X.
(c) What is the variance of the binomial random variable X?
10. An apple farmer named Joe decided to count the number of apples that his 20 newest trees
produced. This is what he discovered:
500, 500, 600, 600, 550, 400, 550, 500, 550, 600
500, 500, 400, 600, 450, 400, 450, 400, 550, 400
(a) Fill in the following table with frequencies and relative frequencies for Joe’s apple production.
Profit (in $) Frequency
400
Relative Frequency
450
500
550
600
650
(b) Draw a histogram representing this data. Be sure to label your axes.
(c) What is the mean number of apples for farmer Joe’s new trees?
(d) What is the median number of apples for farmer Joe’s new trees?
11. (a) What does it mean for two events, E and F , to be independent? Be very clear. Give an
example two events that are independent. Be sure to clearly state the experiment and the
events.
(b) How are the probabilities if E and E 0 related?
(c) What is the definition of variance? If you have two sets of data with the same mean, but
different variance, what does this imply about the data sets?
(d) State the Inclusion/Exclusion principle for Probability.
12. My son is having a snack. In front of him are 4 graham crackers and 3 slices of peach.
(a) He decides to eat two things in front of him. He does this at random. He picks one snack,
eats it, and then picks the next.
i. What is the probability he eats a slice of peach first and a graham cracker second?
ii. What is the probability he eats two slices of peach?
(b) Suppose this time he decides to pick one thing, put it back on the table, and then pick again.
i. What is the probability he picks a slice of peach first and a graham cracker second?
ii. What is the probability he picks a slice of peach both times?
13. There are 20 new songs I would like to download to my phone. 15 are Country songs and 5 are
from Broadway musicals. I only have storage for 7 songs.
(a) What is the size of the sample space here? (i.e. how many ways are there to choose 7 songs.)
(b) How ways are there to choose all Country songs (still only choosing 7 songs total)?
(c) What is the probability that I selected all country songs?
(d) What is the probability that I will select at least one Broadway song?
14. A local club plans to invest $10,000 to host a baseball game. If the team loses the previous game,
they expect to sell tickets worth $9,000. If the team wins the previous game they expect to sell
tickets worth $15,000.The team has a .75 probability of winning any given game, is this a good
investment?
(a) Fill in the table below.
Result of previous game Money Earned
Probability
Win
Lose
(b) What is the expected amount of money they should expect to gain if they host the game?
(i.e. Expected value)
(c) If instead they invest their $10,000 into a spaghetti fundraising dinner, they expect to gain
$3,000. Which should they do?
15. (a) Is the following a feasible probability distribution? Explain why or why not.
Outcome
s1
s2
s3
s4
Probability
.2
0
.1
.8
(b) Let X be a random variable with the probability distribution listed below. Find the probability distribution of 3X 2 − 1.
k
-1
0
1
2
3
P r(X = k)
.2
.1
.4
.1
.2
(c) Which would you expect to have a larger standard deviation, the data set A or the data set
B. Explain your answer.
A = {0, 10, 20, 20, 40, 50}
B = {20, 20, 20, 20, 20, 20, 20, 20}
16. Match the pie charts and bar charts. Explain your answer.
A
30
40 %
20
B
20 %
10 %
10
30 %
C
D
0
A
B
C
D
A
B
C
D
B
25
A
20
20 %
20 %
C
15
20 %
10
40 %
5
0
D
A
4
50 %
3
2
6%
4%
40 %
B
D
C
1
0
A
B
C
D