CSCI 2011 Probability Problems – Numerical Value Questions Hello, Who can help me with questions in the file? Thanks. CSCI 2011, Spring 2019, Assignment 10

Due Wednesday, 4/10/19, 1:00 PM (submission link on Canvas)

This is not a collaborative assignment. You may not show your solution to anyone other than

the course instructor or TAs. In addition, you may not include solutions or portions of

solutions obtained from any source other than those provided in class. Obtaining or sharing

solutions to any assignment for this class is considered academic misconduct. If you are not

sure what this means, consult the class syllabus or discuss it with the course instructor.

Complete the following problems and submit your solutions in a single pdf file to the

Assignment 10 submission link on Canvas. Typed solutions are preferred, but pictures or

scans of a handwritten solutions in pdf form are acceptable so long as your solutions are

clearly legible.

This assignment contains ten problems. Five of these will be graded out of 6 points each; the

remaining five will be assigned 2 points each for completion if it appears that a reasonable

attempt was made. This means that the assignment as a whole is out of 40 points.

Unless otherwise specified, your answers to these problems must be clearly explained in a

step-by-step manner; for most problems, the explanation will be worth far more points than

the actual answer.

To the left of each problem is an odd-numbered textbook problem which the given problem is

based on; these usually have answers given at the back of the textbook. If you dont

understand how to do a given problem, you may want to consider asking a TA (either in office

hours or via email) to explain how to solve the related textbook problem.

Problems 1-7: You must show your work?, and your final answer should be an

approximate numerical value (a decimal or a percentage), not a fraction. Round to three

significant figures. You may want to use scientific notation for some of the extremely unlikely

events.

1. (7.1.25) Suppose that you are playing a lottery which requires you to select 5 distinct

integers between 1 and 20 (order does not matter). Assuming that the 5 winning

numbers are randomly selected, find the probability of the following.

a. Matching all 5 numbers

b. Matching 4 out of the 5 numbers

c. Matching none of the numbers

2. (7.1.35) In terms of credit count, there are 7 freshmen, 77 sophomores, 82 juniors, and

52 seniors in this course, for a total of 218 students. Suppose that we chose five

distinct students at random.

a. What is the probability of choosing five sophomores?

b. What is the probability that at least two of the students chosen have the same

academic standing?

c. What is the probability of choosing at least one junior?

3. (7.1.15) What are the probabilities of getting each of the following in a five-card poker

hand, if only one standard 52-card deck is used?

Notes:

? The order of cards is: 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King, Ace.

? For the sake of simplicity, assume that Ace only appears on the high

end, so even though Ace, 2, 3, 4, 5 is normally a valid straight, we

ignore it.

? For the purposes of this problem, straight flushes (5 cards in a row, all of

the same suit), count as both straights and flushes, so you dont have to

subtract them out for parts (a) and (b).

a. A flush: all 5 cards are the same suit, e.g. all spades (values dont matter)

b. A straight: five values in a row, e.g. (suit doesnt

matter)

c. Two pairs: two cards of one value, two cards of another, different value, and

one card of a third, different value e.g. (suit doesnt

matter)

4. (7.2.25) What is the conditional probability of these events when we randomly select a

permutation of the 10 decimal digits: 0123456789?

a. The permutation is exactly 1357924680, given that the first eight digits are

13579246.

b. The permutations last two digits are 57, in that order, given that the

permutation does not end with 4.

c. The first and last digits of the permutation sum to 11, given that the fourth digit

is 8, and the fifth digit is 9.

5. (7.2.23) What is the conditional probability that flipping a fair coin 6 times will result in

at least 4 heads, given that:

a. The first flip came up heads

b. The last three flips came up tails

c. The first and the last flips were the same

6. (7.3.1) Suppose that E and F are events in a sample space and P(E) = 3/8, P(F) = 1/3,

and P(F | E) = 2/3. Find the following quantities:

a. P(E | F)

b. P(E?C? | F)

c. P(E | F?C?)

7. (7.3.9) Suppose that one in 5000 people is secretly a vampire. There is an excellent

test for vampirism: 99.8% of vampires will spontaneously combust when exposed to

direct sunlight, and only 0.01% of non-vampires will spontaneously combust. What is

the probability that:

a. A person who combusts when exposed to direct sunlight is a vampire?

b. A person who combusts when exposed to direct sunlight is not a vampire?

c. A person who does not combust when exposed to direct sunlight is a vampire?

8. (7.3.19) Suppose that a Bayesian spam filter is trained on a set of 500 spam

messages and 1000 messages that are not spam. The phrase final warning appears

in 200 spam messages and 50 messages that are not spam. Would an incoming

message be rejected as spam if it contains the phrase final warning and the

threshold for rejecting a message is 0.9? Assume, as the textbook does, that the

probability that a random message will be spam is 1/2. Show your work.

9. (7.4.7) There are 48 in-class exercises in this class, total. Suppose that for any given

exercise, the probability that you will submit an answer is 90%, and that if you answer,

the probability that you get the answer correct is 75%. What is the expected number

of total in-class exercises you will get correct? Show your work, and round your final

answer to the nearest integer.

10. (7.4.5) What is the expected value for the following scenarios:

a. Rolling a single 20-sided die (numbered 1-20)

b. The sum of three 6-sided dice

c. The product of two 8-sided dice

Show your work.

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