## Exercises for Tuesday 24 October

Many of these borrowed from Mike Titelbaum.

For the following problems, assume that "a confirms b" (symbolized as aCb) is true for a probability distribution p iff p(b|a) > p(b). Assume that "a disconfirms b" (symbolized as aDb) is true for p iff p(b|a) < p(b).

When giving examples, you may find it helpful to work with a sample space that is a 12-sided die, where all twelve sides are initially regarded as equally likely.

1. Give an example where aCb, bCd, but aDd.

2. Give an example where aCb, aCd, but aD(b or d).

3. Give an example where aCd, bCd, but p(d|ba) = p(d|b), so "b and a" doesn't confirm d any more than b does alone. (This is described as b "screening off" a's confirmatory effect on d.) Assume that b does not entail a.

4. Give an example where a rational agent conditionalizes on new evidence E, yet her credence in a proposition H1 that's consistent with E decreases. That is, H1 does not entail not-E, but p(H1|E) < p(H1).

5. Prove that when an agent conditionalizes on new evidence E, her credence in a proposition H2 that entails E cannot decrease.

6. You're about to roll five fair six-sided dice (as in the game Yahtzee). How confident should you be that there will be no pairs (where there's a pair if there are two or more dice with the same result)?

7. Continuing the previous problem, how confident should you be that there will be no triples?

8. Your friend Heather is addicted to dice games. To help control her addiction, she flips a fair coin each day to decide how much to bet that day. If the coin comes up heads, she bets nothing (but still goes ahead and plays dice games anyway, for fun). If the coin comes up tails, she bets that she'll get at least a triple when tossing five dice. (By "at least a triple," I mean that at least three dice will show the same result. We'll ignore other kinds of rolls that score high in dice games like Yahtzee, such as "straights.") When walking through the department today, you saw Heather emerging from the dice game with a smile on her face. Clearly either Heather bet and won, or she didn't bet and when she rolled the dice anyway they came up bad (no triple). You consider the outcome of the dice tossing to be independent of whether Heather bet.

1. Using a language with the atomic propositions B for "Heather bet" and D for "the dice came up with (at least) a triple", what did you learn when you saw Heather smiling?

2. After updating on the information you just articulated, how confident should you be that the dice came up with (at least) a triple?

3. How confident should you be that Heather bet?

4. Explain why one of the credences you calculated in (b) and (c) matches the value it had prior to you updating on what you learned when you saw Heather smiling, whereas the other differs.

1. Give an example where a is probabilistically independent of b, and b is probabilistically independent of d, but a is not probabilistically independent of d.

2. Give an example where a is probabilistically independent of b, and b is probabilistically independent of d, and a is also probabilistically independent of d, but a is not probabilistically independent of "b and d".

3. Give an example where p(a|b) = p(a|d), but ≠ p(a|b or d).

4. Your friend Alan has credences p(a) = 1/3 and p(b) = 1/3 and p(ab) = 1/6. His p(a|b) is also 1/3. Describe a set of bets that Alan's credences sanction as fair, but which guarantee that he will lose money, once the truths about a and b are settled. Hint: use a conditional bet, where a bet on X conditional on Y pays one side money if X and Y are both true, the other side money when not-X and Y are true, but no money changes hands when Y is false.