If we’ve decided that at the significance level we find comfortable there are too few clean sweeps of any position in Contestants Row, the natural question is why there are so few. We estimated there should have been six clean sweeps, based on modelling clean-sweep occurrences as a binomial distribution. Something in the model went wrong. Let’s try to reason out what it was.
One assumption for a binomial distribution are that we have some trial, some event, which happens many times. Each episodes is the obvious trial here. The outcome we’re interested in seeing has some probability of happening on each trial; there is indeed some probability of a clean sweep each episode. The binomial distribution assumes that this probability is constant for every trial, that it doesn’t become more or less likely the tenth or hundredth or thousandth time around, and this seems likely to hold for The Price Is Right episodes. Granted there is some chance of a clean sweep in one episode; what could be done to increase or decrease the likelihood from episode to episode?
That does suggest a possibility, actually: maybe contestants get better with time. Contestants who’ve seen a particular item come up for bid before can use that knowledge to make a better bid again. Most long-running game shows see the contestants getting better, since contestants tend to be fans and they learn how to play, both faster — in the Chuck Woolery days Wheel of Fortune used to rarely get to a fourth word puzzle during the day; today it averages around seventy completed puzzles per half-hour (admittedly, Woolery’s era lost time to shopping for prizes) — and better, learning strategy from watching other successful contestants (there’s a reason the Wheel of Fortune bonus round just gives every contestant the letters R-S-T-L-N and E). However, it’s hard to see how improved quality of contestants could affect whether one seat gets all the winning bidders. On average each contestant, in each seat, should get better by about the same amount (assuming they do; some weeks, one wonders if contestants do learn). So the probability of a clean sweep in any particular show is probably reasonably constant.
We’ve got the number of hourlong episodes right. Perhaps Drew Carey was simply wrong when he said there had been only one clean sweep before. He wasn’t acting as an archivist, after all, or accountant; his job is to make each show exciting and seeing a rare event is exciting. Bob Barker was — notorious may be literally the correct word, but it sounds ridiculous in this context — reliable, then, for claiming events to be unprecedented when the faithful watcher could determine they were not. And possibly Drew Carey meant to say there had only been one clean sweep while he was hosting; he’s hosted for fewer than a thousand shows, but this would be quite reasonably likely enough. This is a tolerably likely explanation.
But there is also another credible explanation. We supposed — well, I supposed, and only one person spoke up to question this — the chance of all six contestants coming from the same seat would be about one in 1,000, based on the assumption that each of the four seats was equally likely to win each of the six items up for bid. But what if they’re not equally likely? The goal of the item-up-for-bid part is to bid as close as possible to an item’s actual retail price without going over. That’s always going to be won by the person who can name the price exactly right, which could be anyone, but few people manage that. So if nobody knows the exact price, there is an advantage given to the last bidder: she or he can pick whatever sounds like the best of the three given bids, and bid a dollar over that, or bid one dollar if they all sound high, and secure a win. There is, plausibly, an advantage given to the last bidder — which is never the same seat as whoever won the last time around. (Further, a contestant who’s been through several bids can hear if someone else in the audience is reliably correct, which is permitted in the game; the one perfect bid in Showcase history was given by a contestant who paid attention the person in the audience who had memorized the prices of likely prizes.)
In this interpretation, the chance of a clean sweep is less than one in 1,000. It’s hard to guess just what it should be, but it wouldn’t surprise me if it turns out to be one in 6,000. From that we could estimate just how much less likely the first bidder is to win an item up for bid, compared to later bidders.
This may all seem like I’ve taken close to ten thousand words to explain what is obvious: it’s better to be the last bidder in Contestants Row. So I have. But now we have close to five thousand words of justification for that obvious conclusion.