Putting together links to all my essays about trapezoid areas made me realize I also had a string of articles examining that problem of The Price Is Right, with Drew Carey’s claim that only once in the show’s history had all six contestants winning the Item Up For Bids come from the same seat in Contestants’ Row. As with the trapezoid pieces they form a more or less coherent whole, so, let me make it easy for people searching the web for the likelihood of clean sweeps or of perfect games on The Price Is Right to find my thoughts.
Came On Down: This was the essay which introduced the question. Drew Carey made the claim that only once in, by my estimate, six thousand episodes of the hourlong show had all six contestants come from the same seat. A bit of reasonable guessing based on how often contestants from each seat might win indicated that such a clean sweep should have happened about once every thousand episodes, or about six times by now.
From Drew Carey to an Imaginary Baseball Player introduces the binomial distribution, and a simple question: just because something might be expected to happen one time out of a thousand attempts, does it follow that in 6,000 attempts it should happen exactly six times? Could’t it happen more or fewer times?
Off By A Factor Of 720 (Or More) accepts that, certainly, something happening about once out of every thousand attempts might happen once, or twice, or thrice, or more in six thousand attempts. To estimate just how likely each number of occurrences is we need to look into combinations, into all the ways that a set number of occurrences could come up in a given number of tries.
A Simple Demonstration Which Does Not Clarify lives up to its promise: I show how to work out the likelihood of zero, or one, or two, or all the way up to fifteen clean-sweep episodes in 6,000 shows. But knowing just how likely it is to have, say, six clean sweeps in 6,000 shows doesn’t by itself tell us whether there’s something peculiar about there being just the one.
Significance Intrudes on Contestants Row: Here we more explicitly introduce the idea of significance. Extremely unlikely things can and do happen; why should we attach significance to that? For that matter, when should we start attaching significance to that?
The First Tail provides a provisional answer. With a binomial distribution like this we can say how likely or unlikely it is that there were as few clean sweep episodes as we heard reported. This is a measure not just of the probability of having a single clean sweep, but of having no more than one clean sweep.
The Significance Of The Item Up For Bids puts forth the idea that we can draw a line. We can say that we consider some threshold of likelihood, and say that outcomes which are less likely than that are probably significant. We may have fewer clean sweeps than we expected, but do we have so few that it’s implausible to say it was just a matter of chance?
Finding, And Starting To Understand, The Answer: what’s hard about this is that there’s not one absolute standard about how improbable an outcome has to be before we find it suspicious. There are several common threshold levels, based on our fondness for numbers that are nice and round in base ten, and it turns out that picking different thresholds gives us an answer of yes or no, depending.
Interpreting Drew Carey: If we picked a threshold for significance which indicated that there are just suspiciously few clean sweeps, then we’re left with the mystery, why are there so few? A couple of potential answers are given, and I think they’re all defensible.
Figuring Out The Penalty Of Going First: There is one explanation for this deficiency which feels quite plausible, though. That’s to suppose that some bidders on Contestants’ Row are at a relative advantage or, maybe more important, at a disadvantage. Watching The Price Is Right suggests that being the last contestant to bid is probably the choice spot. Being the first contestant to bid is probably the worst. If we assume that one clean sweep in six thousand episodes is the most likely outcome, this lets us say at least how big a penalty being the first bidder probably carries.
What Can One Week Prove? If we’ve supposed there to be a penalty to going first, a natural follow-up question is, can we prove it? How many Items Up For Bid would we have to watch to be able to say whether going first probably does hurt one’s chances of winning?
Proving Something With One Month’s Counting: One week’s worth of Items Up For Bid — there are six each episode, after all, and five episodes per normal week — would give us some idea of whether the first bidder is at a disadvantage, but wouldn’t demonstrate it to statistical significance. But in surprisingly little time we could gather enough data to say whether the first bidder’s disadvantage is high enough to explain the lack of clean sweeps. I haven’t actually gathered this data, and as far as I know nobody else has either. But we know how much to look for, at least.
So If You Can’t Win The Clock Game You Should Feel Bad rounds out the subject by trying a slightly different question. There have been, allegedly, 80 perfect episodes, ones in which all six contestants win the pricing games played after they come up on stage. Given that, then, what is the average likelihood of any contestant winning any particular pricing game? Now, some pricing games are easy (two are even winnable on pure skill without any pricing knowledge or luck); some might as well be impossible. But we can take an average which means something, surely, to someone. And from just these scanty pieces of information — the number of episodes, the number of perfect episodes, that there are six pricing games per episode — we can say, what is the likelihood of winning a Price Is Right pricing game?
There’s a lot of handwaving in this argument, but all the steps are reasonable enough, and the answer feels about right.
Not done, but easy enough to work out using the tools introduced, would be figuring out how many episodes, how many pricing games, would have to be observed to say whether the claimed probability of winning a pricing game was likely correct. Let’s consider that an exercise for the interested student.