Probability on Jeopardy!

No, this isn't about the probability of winning at Jeopardy or any such thing.

On Wednesday night's Jeopardy!, there was a category "Fun With Probability". It's always funny to see math categories appear on the show, because the contestants seem to avoid them. (They shy away from science categories as well, but not to the same extent; you can fake your way through a science category by having memorized a bunch of things, but you can't do that with a math category. By the way, any puns on the word "category" I might make in this post are unintentional.)

You can see the entire game at the Jeopardy! archive. The questions were as follows:

$400: High rollers get to play roulette on single-zero wheels, where the chance of hitting your lucky number is 1 in this

Contestants answered "36" and "32". I think "32" was a wild guess. But "36" is actually a fairly reasonable answer here. The odds of hitting your lucky number on a single-zero wheel are, in fact, 36 to 1. A single-zero wheel has thirty-seven spots -- the zero and the numbers one through 36 -- and a bet of $1 pays $36 if your number comes up. The house edge here is just 1/37, as opposed to 2/38 on the wheels with both zero and double-zero. (Is there a form of roulette where there are no zeroes? This seems like it could exist if not played in a casino. Poker is a fair game when played socially but in casinos the house takes a percentage of each pot. Then again, I don't see people getting together socially to spin a wheel and hand each other money; poker involves infinitely more skill than roulette. I mean the word "infinitely" here literally, because roulette takes zero skill.)

$800: The chance of getting heads on any given coin flip is 1 in 2, so the chance of getting heads 5 times in a row is 1 in this

One in thirty-two, of course; thirty-two is 2⁵. (Trick question that I might give if I ever find myself teaching basic probability: the odds of getting heads on a given coin flip of a certain unfair coin are 2 to 1 against. What are the odds of getting heads five times in a row? The answer is 242 to 1 against, but I suspect a lot of people would hear "coin" and just start multiplying out the twos.)

$1200: NASA's Spaceguard Survey watches for the "extremely small" probability of these 2 objects coming to smash Earth

Not really a probability question; you really just had to kind of guess your way through this one. There are three reasonable objects -- comets, meteors, and asteroids. The three contestants, in turn, guessed all three possible pairs of them; binomial coefficients in action! (There are plenty of Jeopardy! clues where there are three possible answers, and the best strategy seems to be to wait for the other two people to get the wrong answer. There are also a fairly large number where there are two possible answers; Sweden/Norway and Oxford/Cambridge seem like common pairs of this sort. At least for me.) If you know a bit of astronomy, though, you know that meteors are not that big, and hit the earth all the time in meteor showers.

$1600: Offspring of heterozygous parents have a 50-50 chance of getting a dominant vs. this type of allele

A lot of Jeopardy! clues have a bunch of extraneous verbiage; the question here is really "what's the kind of allele that's not dominant?", which might be surprisingly easy for a $1600 clue. But at this point I feel obliged to mention that genetics is one of the other disciplines where probability and statistics were applied early on, or so a friend of mine tells me; I suppose this is more reputable than gambling and less boring than insurance.

$2000: The probability of the first card dealt being an ace is 4 in 52, so the probability of the second card being an ace is 1 in this number

(There was a video here; it was pretty clear that this was the probability of the second card being an ace, given that the first card also was an ace.) There are three aces left among the fifty-one remaining cards, so it's three in 51, which is one in 17. I was screaming at the TV -- none of the contestants got it -- but then again I do this stuff for a living, and as far as I know none of them do, so I really shouldn't be too hard on them. Incidentally, this is a "baby version" (as one of my the principle at work in card counting in blackjack; if you know the deck is rich in high cards then the dealer is more likely to bust, so you bet more.