probability question need in 10 hrs
enggsd
assignment3.pdf
COMP 2804 — Assignment 3
Due: March 20, before 23:59 pm, in the course drop box in Herzberg 3115.
Assignment Policy: Late assignments will not be accepted. Students are encouraged to collaborate on assignments, but at the level of discussion only. When writing the solutions, they should do so in their own words. Past experience has shown conclusively that those who do not put adequate effort into the assignments do not learn the material and have a probability near 1 of doing poorly on the exams.
Important note: When writing your solutions, you must follow the guidelines below.
• The answers should be concise, clear and neat.
• When presenting proofs, every step should be justified.
• Assignments should be stapled or placed in an unsealed envelope.
Substantial departures from the above guidelines will not be graded.
If you submit your solutions using LATEX, you will get one bonus mark.
Question 1: On the first page of your assignment, write your name and student number.
Question 2: Consider the sample space S = {a, b, c, d} and a probability function Pr : S → R on S. Define the events A = {a}, B = {a, b}, C = {a, b, c}, and D = {b, d}. You are given that Pr(A) = 1/10, Pr(B) = 1/2, and Pr(C) = 7/10.
What is Pr(D)? Justify your answer.
Question 3: The Fibonacci numbers are defined as follows: f0 = 0, f1 = 1, and fn = fn−1 + fn−2 for n ≥ 2.
Let n be a large integer. A Fibonacci die is a die that has fn faces. Such a die is fair: If we roll it, each face is on top with the same probability 1/fn. There are three different types of Fibonacci dice:
• D1: fn−2 of its faces show the number 1 and the other fn−1 faces show the number 4.
• D2: Each face shows the number 3.
• D3: fn−2 of its faces show the number 5 and the other fn−1 faces show the number 2.
Assume we roll each of D1, D2, and D3 once, independently of each other. Let R1, R2, and R3 be the numbers on the top face of D1, D2, and D3, respectively. Determine
Pr(R1 > R2),
and Pr(R2 > R3),
1
and show that
Pr(R3 > R1) = fn−2fn+1
f2n .
Question 4: You are doing two projects P and Q. The probability that project P is successful is equal to 2/3 and the probability that project Q is successful is equal to 4/5. Whether or not these two projects are successful are independent of each other.
What is the probability that both P and Q are not successful? Justify your answer.
Question 5: According to Statistics Canada, a random person in Canada has
• a probability of 4/5 to live to at least 70 years old and
• a probability of 1/2 to live to at least 80 years old. John (a random person in Canada) has just celebrated his 70-th birthday. What is the probability that John will celebrate his 80-th birthday? Justify your answer.
Question 6: We take a uniformly random permutation of a standard deck of 52 cards, so that each permutation has a probability of 1/52!. Define the following events:
• A = “the top card is an Ace”,
• B = “the bottom card is the Ace of spades”,
• C = “the bottom card is the Queen of spades”. Determine
Pr(A | B), and
Pr(A | C).
Question 7: A hand of 5 cards is chosen uniformly at random from a standard deck of 52 cards. Define the event
A = “the hand has at least one Ace”.
• Explain what is wrong with the following argument:
We are going to determine Pr(A). Event A states that the hand has at least one Ace. By symmetry, we may assume that A is the event that the hand has the Ace of spades. Since there are
( 52 5
) hands of five cards and exactly
( 51 4
) of them
contain the Ace of spades, it follows that
Pr(A) =
( 51 4
)( 52 5
) = 5 52
.
2
• Explain what is wrong with the following argument:
We are going to determine Pr(A) using the Law of Total Probability. For each x ∈{♠,♥,♣,♦}, we define the event
Bx = “the hand has the Ace of suit x”.
We observe that
Pr(Bx) =
( 51 4
)( 52 5
) = 5 52
.
We next observe that Pr(A | Bx) = 1,
because if event Bx occurs, then event A also occurs. Thus, using the Law of Total Probability, we get
Pr(A) = ∑ x
Pr(A | Bx) · Pr(Bx)
= ∑ x
1 · Pr(Bx)
= ∑ x
5
52
= 4 · 5
52
= 5
13 .
• Determine the value of Pr(A). Justify your answer.
Question 8: We are given a tetrahedron, which is a die with four faces. Each of these faces has one of the bitstrings 110, 101, 011, and 000 written on it. Different faces have different bitstrings.
We roll the tetrahedron so that each face is at the bottom with equal probability 1/4. For k = 1, 2, 3, define the event
Ak = “the bitstring written on the bottom face has 0 at position k”.
For example, if the bitstring at the bottom face is 101, then A1 is false, A2 is true, and A3 is false.
• Are the events A1 and A2 independent? Justify your answer.
• Are the events A1 and A3 independent? Justify your answer.
3
• Are the events A2 and A3 independent? Justify your answer.
• Are the events A1, A2, A3 pairwise independent? Justify your answer.
• Are the events A1, A2, A3 mutually independent? Justify your answer.
Question 9: In a group of 100 children, 34 are boys and 66 are girls. You are given the following information about the girls:
• Each girl has green eyes or is blond or is left-handed.
• 20 of the girls have green eyes.
• 40 of the girls are blond.
• 50 of the girls are left-handed.
• 10 of the girls have green eyes and are blond.
• 14 of the girls have green eyes and are left-handed.
• 4 of the girls have green eyes, are blond, and are left-handed.
We choose one of these 100 children uniformly at random. Define the events
G = “the kid chosen is a girl with green eyes”,
B = “the kid chosen is a blond girl”,
and L = “the kid chosen is a left-handed girl”.
• Are the events G and B independent? Justify your answer.
• Are the events G and L independent? Justify your answer.
• Are the events B and L independent? Justify your answer.
• Verify whether or not the following equation holds:
Pr(G∧B ∧L) = Pr(G) · Pr(B) · Pr(L).
Hint: Draw a Venn diagram.
Question 10: By flipping a fair coin repeatedly and independently, we obtain a sequence of H’s and T ’s. We stop flipping the coin as soon as the sequence contains either HH or TH.
4
Two players play a game, in which Player 1 wins if the last two symbols in the sequence are HH. Otherwise, the last two symbols in the sequence are TH, in which case Player 2 wins. Define the events
A = “Player 1 wins”
and B = “Player 2 wins.”
Determine Pr(A) and Pr(B). Justify your answer.
5
