Solving The Price Is Right’s “Any Number” Game
A friend who’s also into The Price Is Right claimed to have noticed something peculiar about the “Any Number” game. Let me give context before the peculiarity.
This pricing game is the show’s oldest — it was actually the first one played when the current series began in 1972, and also the first pricing game won — and it’s got a wonderful simplicity: four digits from the price of a car (the first digit, nearly invariably a 1 or a 2, is given to the contestant and not part of the game), three digits from the price of a decent but mid-range prize, and three digits from a “piggy bank” worth up to $9.87 are concealed. The contestant guesses digits from zero through nine inclusive, and they’re revealed in the three prices. The contestant wins whichever prize has its price fully revealed first. This is a steadily popular game, and one of the rare Price games which guarantees the contestant wins something.
A couple things probably stand out. The first is that if you’re very lucky (or unlucky) you can win with as few as three digits called, although it might be the piggy bank for a measly twelve cents. (Past producers have said they’d never let the piggy bank hold less than $1.02, which still qualifies as “technically something”.) The other is that no matter how bad you are, you can’t take more than eight digits to win something, though it might still be the piggy bank.
What my friend claimed to notice was that these “Any Number” games went on to the last possible digit “all the time”, and he wanted to know, why?
My first reaction was: “all” the time? Well, at least it happened an awful lot of the time. But I couldn’t think of a particular reason that they should so often take the full eight digits needed, or whether they actually did; it’s extremely easy to fool yourself about how often events happen when there’s a complicated possibile set of events. But stipulating that eight digits were often needed, then, why should they be needed? (For that matter, trusting the game not to be rigged — and United States televised game shows are by legend extremely sensitive to charges of rigging — how could they be needed?) Could I explain why this happened? And he asked again, enough times that I got curious myself.
Estimating how often the game should take eight digits to solve — let me call it the Suspenseful case — struck me as practically impossible to do by reason alone. Some little problems, like working out the probability of sweeping the piggy bank or mid-range prize or the car, are certainly doable and might make decent Intro to Probability questions. But it seemed to me there are so many ways to select seven digits out of ten without getting a prize that I’d become trapped in the labyrinth of subcases trying to work that out.
With reasoning out as being too much work, empirical estimates start looking good. We could watch The Price Is Right an awful lot, pay attention each time “Any Number” comes up, and count how many digits it takes to win a prize. I suspect we’d probably need a couple hundred rounds to get anything a little bit plausible; if “Any Number” is played one or two times a week, then, that’s a couple seasons of watching before we can say something. I don’t actually know that it’s played one or two times a week; that just feels like about how often it turns up, and I know what I just said about impressions. But it’s a popular game, it doesn’t require any complicated props, and it seems (again with the unexamined impressions) to take about the same amount of time to play each game, so it should be very attractive to the people who schedule the show’s games.
With the actual show being out as too much work, we would seem to be stuck. This is what computers are for: what I could do is model the playing of the game and then run a whole bunch of games, and see how often it goes to the Suspenseful case. So, I wrote a little function in Octave (an open-source clone of Matlab, which does numerical computing and plotting quite well) which simulated the contestant picking digits, figures how long it takes to win a prize, and reports how many digits were picked and what prize was won. (I might talk about how the details of the model were chosen later.)
There are in principle — well, let’s leave that for a homework exercise — quite a few ways to select eight digits out of the ten possibilities. It’s a big number, but it’s not one so enormous that a laptop computer like mine can’t do it without even requiring its full attention. I tried running it with a bunch of likely-looking numbers, to get an idea of how frequently each outcome came up. Here’s the results from as few as a hundred to as many as ten million simulated games; I could fill in more rows but that would be crushingly boring as you can see the Law of Large Numbers take full effect:
|Prize Takes (N) Digits||3||4||5||6||7||8|
|In 100 Games||0||3||12||25||31||29|
|In 10,000 Games||164||525||1,155||2,041||3,087||3,028|
|In 10,000,000 Games||166,390||547,691||1,189,033||2,094,016||3,002,429||3,000,441|
So my friend has a perfectly good reason to think the game takes to the last digit “all the time”: it actually does, three times out of ten. It takes to the 7th or 8th digit about three-fifths of the time, which is probably why the game seems to take such a steady amount of screen time. It also suggests the game really does naturally tend towards the Suspenseful case, which might be part of what makes it popular among viewers.
And if you were curious, the number of times the car, the mid-range prize, and the piggy bank are won are just what you might have guessed, at least qualitatively. Here let me just use the breakdown from the set of 10,000,000 simulated games, which suggest that while the car is given away a comfortable quarter of the time, the contestant will on average have winnings worth a couple hundred dollars, which might be part of what makes it popular among (I imagine) the show’s budget directors:
|Prize||Car||Mid-Range Prize||Piggy Bank|
Now, some weaknesses to this: this little model assumes that contestants are guessing randomly, that the contestant has absolutely no idea what numbers might possibly go where. A dedicated Price Is Right fan will probably say this is perfectly correct, based on the number of people who can’t get the hang of “Clock Game”, and who haven’t noticed how to crack “Cliffhangers”. (I won’t spoil it, but if you’re figuring to be on the show, wouldn’t you look for tips on the games that you can strategize for?) But there’s no actual guarantee that contestants are guessing randomly. A sharp contestant would be able to make fairly good guesses about what the car’s prices are, and throw these percentages off.
Of course, it’d take a long round of watching the show and counting the results to tell whether contestants were, on average, better or worse than random chance. You can try counting.
(Or just mining the past data: the good people at Golden-Road.net have daily recaps of the episodes in the sort of detail that baseball statistics enthusiasts would understand. Some commenters even keep track of how long the games take to play.)