On the December 15th episode of The Price Is Right, host Drew Carey mentioned as the sixth Item Up For Bids began that so far that show, all the contestants who won their Item Up For Bids (and so got on-stage for the pricing games) had come from the same spot so far, five out of six. He said that only once before on the show had all the contestants come from the same seat in Contestants Row. That seems awfully few, but, how many should there be?
We can say roughly how many “clean sweep” shows we should expect. There’ve been just about 6,000 episodes of The Price Is Right played in the current hour-long format (the show was a half-hour its first few years after being revived in 1972; it was a very different show in previous decades). If we know the probability of all six contestants in one game winning their Item Up For Bids — properly speaking, it’s called the One-Bid, but nobody cares — and multiply the probability of six contestants in one show coming from the same seat by the number of shows, we have the number of shows we should expect to have had such a clean sweep. This product, the chance of something happening times the number of times it could happen, is termed the “expected value” or “expectation value”, or sometimes just the “mean”, as in the average number to be, well, expected.
This makes a couple of assumptions. All probability problems do. For example, it assumes the chance of a clean sweep in one show is unaffected by clean sweeps in other shows. That is, if everyone in the red seat won on Thursday, that wouldn’t make everyone in the blue seat winning Friday more or less likely. That condition is termed “independence”, and it is frequently relied upon to make probability problems work out. Unfortunately, it’s often hard to prove: how do you prove that one thing happening doesn’t affect the other?