4.1. A bag has 4 coins in it: 2 are fair coins and two are loaded coins. Each loaded coin comes up heads with probability 3/1. If we select two coins uniformly at random from this bag and flip them, compute the probability we get two heads.

4.2. If Taylor studies very hard there’s a 90% chance she will ace the exam. If she doesn’t study very hard, there’s only a 40% chance she’ll ace the exam. Past experience dictates that Taylors studies very hard for exams 20% of the time. If Taylor doesn’t ace the exam, what’s the probability Taylor studied very hard for it?

4.3. Three independent events A, B and C occur with probabilities P(A) = .5, P(B) = .75, and P(C) = .2, respectively.

(a) If at least one of the three events occurs, what is the probability that only A occurs?

(b) If exactly one of the three events occurs, what is the probability that A occurs?

4.4. (a) There is another notion of independence called conditional independence. We say A and B are conditionally independent given C provided P(A∩B|C) = P(A|C)P(B|C). That is, A and B are inde-pendent with respect to the probability law P(·|C) (and not necessarily the unconditional probability P(·)). Suppose A and B are conditionally independent given C. Show that P(A|B ∩ C) = P(A|C). You may assume P(C) > 0 and P(B|C) > 0.

(b) A box has two coins (one fair, the other comes up heads with probability 3/1). You grab a coin uniformly at random and flip it twice. Let H1 (resp., H2) be the event that the first (resp., second) toss shows heads. Show that P(H1 ∩ H2) = P(H1)P(H2) – so that these events are not independent.

Remark. Note that if F is the event you pick the fair coin, then P(H1 ∩ H2|F) = P(H1|F)P(H2|F) and P(H1∩H2|F c ) = P(H1|F c )P(H2|F c ) and H1 and H2 are conditionally independent given we know which coin we selected.

4.5. You roll a die repeatedly. If at least one 5 occurs before the first 6, what’s the probability the 6 happens on the third roll? What’s the probability that the first six occurs before the first 5?

4.6. An urn has 4 balls: 1 white, 1 green and 2 red. We draw 3 balls with replacement. Find the probability we did not see all three colors. Try answering this two ways: one way using inclusion exclusion, another way by considering the complement event.

4.7. There are k people who are going to draw chips from a box containing n chips numbered 1,2,3,. . . ,n. Each person selects r chips without replacement, notes their selection, and, when they are done, they replace the chips back into the box for the next person until all k have made their selections.

(a) What’s the probability all k selections have at least one chip number in common?†

(b) Now, let 0 < m ≤ r and suppose the box has m special chips. Write an expression for the probability that all k people drew the special chips.

(c) (continued from (b)) Write an expression for the probability that none of the people drew the special chips.

† Hint: If we define Ci to be the event that all k people select chip number i, then ∪n i=1 Ci is the event that there is at least one chip number that all k people select.

4.8.* (a) Each day (5 days per week, 52 weeks per year) Lucy buys a $5 Scratch-off Lottery ticket that claims there’s a .0001 chance of winning the jackpot of $10,000. If we let Wi be the event that Lucy wins the jackpot on the ith lottery ticket, compute the probability that Lucy wins the jackpot (at least once) in 8 years of playing. How much money would she have spent buying the lottery tickets for these 8 years? and, how does this compare with the jackpot?

(b) How many days would Lucy need so that the probability she wins the jackpot at least once in her lifetime is at least 50%?

* You’ll need a calculator to ultimately compute the values in this problem. You’ll also need to make an assumption about the events W1, W2, W3, . . . . Assume Lucy will continue playing this lottery each day regardless of whether she wins or loses in previous days.

4.9. (a) Explain why (2n n) = ∑n k=0 (n k)2.

(b) A and B toss a coin n times. Compute the probability they each toss the same number of heads.

(c) (continued from part (b)) By symmetry, the probability that A tosses fewer heads than B and the probability that A tosses more heads than B are the same, call this common probability p. Use this fact and part (b) to compute the value of p.

(d) A tosses a coin n + 1 times and B tosses a coin n times. A wins if A tosses more heads than B. Compute the probability that A wins by conditioning on the state of the game after A and B have each tossed their coins n times.