1. (Tree Diagrams) Wayne and Kai keep playing chess until one of them wins two games in a row or one of them wins three games (not necessarily in a row).
(a) In what percentage of all possible cases does the game end because Wayne wins three games without winning two in a row?
(b) Supposing Wayne and Kai are equally skilled at chess, so that the probability of one of them winning a particular game is 0.5. What is the probability that the game ends because Wayne wins three games without winning two in a row?
(Note that the answer to the two questions is not the same: (a) asks for the percentage of all outcomes, as if all outcomes are all equally likely; but in fact in (b) they are NOT equally likely, so you’ll need to attach probabilities to the branches. Do the best you can to show this tree, it doesn’t have to be pretty, just clear; and then give the answer to the questions.)
2. (Tree Diagrams) Kai has $4. He plays a game with Wayne in which he bets $1 on the ﬂip of a fair coin: if the coin lands heads, Wayne gives him $1 and if tails, he gives Wayne $1. He decides to play this game four times.
(a) Draw a tree diagram showing all the possible outcomes after four games, showing the amount of money Kai has after each game (no need to make this pretty, just make it clear, in whatever format you are submitting the homework).
(b) What is the probability that Kai breaks even (has $4 at the end of four games)?
(c) What is the probability that Kai wins money (has more than $4 at the end)? The next three problems have to do with the random experiment of rolling two fair dice and counting the number of dots that show on both.
3. (Conditional Probability) What is the probability of a sum of 5 if
(a) the second roll is not 3?
(b) they land on diﬀerent numbers?
4. (Conditional Probability) Suppose that the total of the dots showing is found to be divisible by 5. What is the probability that both of them have shown 5 dots?
(a) Let A = "the sum of the dots showing on the two rolls is odd" and B = "both tosses were greater than 3." Are A and B independent? Be precise.
(b) Let A = "the two rolls showed the same number" and B = "the second toss was greater than 4." Are A and B independent? Be precise.
6. (Independence) We assumed in class that the relation of Independence is symmetric, that is, A and B are independent iﬀ (if and only if) B and A are independent. Justify this assumption by proving that P(A|B) = P(A) iﬀ P(B|A) = P(B).
(a) Prove that P(A) = P(AB) + P(AB’). (B’ is the complement of B)
(b) Prove that if A and B are independent, then so are A and B’.
(c) Prove that if A and B are independent, then so are A’ and B’.
(Hint: use (a) to prove (b); (c) is a simply corollary of (b).)
8. (Bayes Rule) A stack of cards consists of 6 red and 5 black cards. A second stack of cards consists of 9 red cards. A stack is
selected at random (i.e., 0.5 probability for each) and a card is drawn and found to be red. What is the probability that it was
drawn from the ﬁrst stack? You must explicitly use Bayes’ Rule to solve this.
9. (Bayes Rule) A faulty communication line for digital signals changes 1/4 of the 0′s to 1′s and 1/3 of the 1′s to 0′s. If 40% of a particular ﬁle being transmitted consists of 0′s and 60% consists of 1′s, what is the probability that when a 0 is received it was transmitted correctly (i.e., a 0 was transmitted)? You must ﬁrst draw a diagram and then explicitly use Bayes Rule to arrive at the solution.
10. (Based on a true story, more or less….)
Wayne falls asleep watching an old Clint Eastwood movie at 4am after preparing his lecture for CS 237, and has a nightmare that goes like this:
He is lying on the sidewalk after robbing a bank, in pain and mulling over how to quantify the uncertainty of his survival, when Dirty Harry walks over. Dirty Harry pulls out his 44 Magnum and puts two bullets opposite each other in the six slots in the cylinder (e.g., if you number them 1 .. 6 clockwise, he puts them in 1 and 4), spins the cylinder randomly, and, saying “The question is, are you feeling lucky, probabalistically speaking, computer science punk?” points it at Wayne’s head and pulls the trigger…. “CLICK!” goes the gun (no bullet) and Dirty Harry smiles… “How about that …. Let’s see if this gun is memoryless!”
Without spinning the cylinder again, he points the gun at Wayne’s head and pulls the trigger again.
(a) What is the probability that (at least in my dream) you will have to have another instructor finish out CS 237?
(b) Now, suppose that when Dirty Harry put the bullets in the gun, he put them right next to each other (e.g., in slots 1
and 2). What is the probability in this case that you will have another instructor finish teaching CS 237?
(c) Suppose Dirty Harry puts the bullets in two random positions in the cylinder and we don’t have any idea where they
are. Now what is the probability that I will not be able to finish teaching CS 237?
Hint: This has nothing to do with the memory-less property (Dirty Harry never took CS 237) …. and we could solve it by just
considering what the probability is for each of the patterns (no bullet, no bullet) and (no bullet, bullet) as we go around the circle of slots in the cylinder.
Here is what the cylinder would look like in case (a) before Dirty Harry randomizes the positions by spinning it: