Question 2: Consider the sample space S = {a, b, c, d} and a probability function Pr: S —> IR on S. Define the events A = {a}, B = {a, b}, C = {a, b, c}, and D = fb, cll. 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: fo = 0, fi = 1, and fn = fn-i + fri_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_i 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_i 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(Ri > R2),
and
Pr(R2 > R3),
Question: 4: You are doing two projects P and Q. The probability that project P issuccessful 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: that each permutation has a probability of 1=52!. De ne 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 8: 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. Di erent faces have di erent bitstrings.
We roll the tetrahedron so that each face is at the bottom with equal probability 1/4.
For k = 1; 2; 3, de ne 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.
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. De ne 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):
Question 10: Question 10: By ipping a fair coin repeatedly and independently, we obtain a sequence of H's and T's. We stop ipping the coin as soon as the sequence contains either HH or T H.
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 T H, in which case Player 2 wins. De ne the events
A = "Player 1 wins"
and
B = "Player 2 wins."
Determine Pr(A) and Pr(B). Justify your answer.