The Mathematics of Pricing Games on The Price is Right

[Jeremy Nelson]

The math behind the Pay The Rent game always fascinates me. Correct me if I'm wrong, but I believe it is, by far, the easiest game to win $10K on the show. However, usually only one combo of items/levels would get you the $100K.

Years ago, I created a rudimentary Pay the Rent analyzer with Excel.  I run the prices through it every time I see the game.  Usually, there are 2-3 possible solutions, but the number has as high as 7.

The original brilliance of that game was that it was counter-intuitive.  A player's instinct is to place the cheapest item at the bottom (necessary), then the next two cheapest on the floor above, the next two on the floor above that, and then the highest priced item in the attic (also necessary).  But (at least originally), the game was set so that the second and third highest priced items added together were more than the highest priced one.  So your middle four had to be carefully chosen (or lucky guesses).  That's why it was worth $100,000!

I'm not super thrilled by the easy setups, but I imagine that it had to be in their plan all along that they would/could alter the setup to produce a winner at some point. Considering that contestants playing Ten Chances still don't understand the zero rule after 40 years of that being a constant, I highly doubt that few, if any contestants that get up on stage will win Pay the Rent as intended.

Actually, now that I'm typing this, I have another question- statistically speaking, assuming the contestant is making random guesses during whatever game they play, is Pay The Rent the only game in the lineup whose win probability can be altered by the producers without changing the rules of the game?

[nowhammies10]

Actually, now that I'm typing this, I have another question- statistically speaking, assuming the contestant is making random guesses during whatever game they play, is Pay The Rent the only game in the lineup whose win probability can be altered by the producers without changing the rules of the game?

Lucky $even immediately springs to mind as another such game.  Contestants tend to stick towards the middle in their guesses; a car priced at $19,191 would virtually guarantee a loss after the second guess.

[Kevin Prather]

Actually, now that I'm typing this, I have another question- statistically speaking, assuming the contestant is making random guesses during whatever game they play, is Pay The Rent the only game in the lineup whose win probability can be altered by the producers without changing the rules of the game?

Lucky $even immediately springs to mind as another such game.  Contestants tend to stick towards the middle in their guesses; a car priced at $19,191 would virtually guarantee a loss after the second guess.

Ten Chances to a lesser extent. A $23,980 car would be easier to win than a $24,560 car.

[TLEberle]

Actually, now that I'm typing this, I have another question- statistically speaking, assuming the contestant is making random guesses during whatever game they play, is Pay The Rent the only game in the lineup whose win probability can be altered by the producers without changing the rules of the game?

Lucky $even immediately springs to mind as another such game.  Contestants tend to stick towards the middle in their guesses; a car priced at $19,191 would virtually guarantee a loss after the second guess.

But that doesn't really change the probability--it's just a hose-job, similar to if Switcheroo's small items don't look to fit a certain value frame the contestant still has a one-in-five chance of getting the car randomly.

[Kevin Prather]

But that doesn't really change the probability--it's just a hose-job

It does change the probability. If a 5 comes up, the most you can miss it by is 5. If a 0 or 9 comes up, you can miss it by as much as 9.

Someone guessing numbers completely at random will lose an average of $2.50 every time a 4 or 5 comes up, $2.70 on a 3 or 6, $3.10 on a $2 or $7, $3.70 on a 1 or 8, and $4.50 on a 0 or 9.

Interestingly, even if the price is $25,555, a person picking numbers at random will still lose the game more often than they would win (although anyone playing with even a modicum of strategy would win every time).

[Steve Gavazzi]

Haven't been watching the show as much as I used to, but it seems like Pocket Change is proving difficult for contestants lately. I haven't seen the $2.00 card come up, but even if it did, that doesn't always guarantee a win.

There's only one $2 card, so not seeing it should hardly be surprising.

As for it not always guaranteeing a win, yeah, that's technically true, but in practice, it's possible to make as many as 10 mistakes while playing Pocket ¢hange, and the $2 card will auto-win the game if you make fewer than nine of them.  In the 15 years it's been on the show, I think only one contestant has manged to play it that badly.

(And yeah, that's not really math, but Mitch's observation isn't really math, either.)

[Jeremy Nelson]

Actually, now that I'm typing this, I have another question- statistically speaking, assuming the contestant is making random guesses during whatever game they play, is Pay The Rent the only game in the lineup whose win probability can be altered by the producers without changing the rules of the game?

Lucky $even immediately springs to mind as another such game.  Contestants tend to stick towards the middle in their guesses; a car priced at $19,191 would virtually guarantee a loss after the second guess.

Ten Chances to a lesser extent. A $23,980 car would be easier to win than a $24,560 car.

Not really- there are still the same number of possible number combinations for both of those prices. Contestants picking random number jumbles using those five have the same win/loss probability. 

[Joe Mello]

Bringing back Dice Game, if the car price only has 2 through 5, the player has a 1/16 chance of winning the car without making a guess*. If the car only has 1's and/or 6's, that probability jumps to 1/81.

[Mr. Armadillo]

Actually, now that I'm typing this, I have another question- statistically speaking, assuming the contestant is making random guesses during whatever game they play, is Pay The Rent the only game in the lineup whose win probability can be altered by the producers without changing the rules of the game?

Going down the list, I only came up with one more - Magic #.

You could make a case for games like Cliff Hangers, Spelling Bee, and Rat Race, I suppose - if you make the products, say, $300 each, nobody's going to win anything, but within reason, I don't think they count.

[Joe Mello]

Actually, now that I'm typing this, I have another question- statistically speaking, assuming the contestant is making random guesses during whatever game they play, is Pay The Rent the only game in the lineup whose win probability can be altered by the producers without changing the rules of the game?

Games like Squeeze Play, Card Game, and 3 Strikes can have different win chances depending on the prize at stake.

[nowhammies10]

Another thought: Stack The Deck?  If the first digit of the car's price isn't immediately obvious, even if the player does everything right and selects the 3rd-5th numbers, leaving 1-2-8-9 as the last four numbers on the line would give 24 (4!) combinations, as opposed to something like 2-5-8-9, which would leave only three "obvious" choices.

[Mr. Armadillo]

The qualifier was "guessing randomly", so anything that requires literally any mental judgement is out.

Games like Squeeze Play, Card Game, and 3 Strikes can have different win chances depending on the prize at stake.

I'd argue that increasing the contestant's choices by 1 counts as a rule change for Squeeze Play and 3 Strikes.

[Unrealtor]

Actually, now that I'm typing this, I have another question- statistically speaking, assuming the contestant is making random guesses during whatever game they play, is Pay The Rent the only game in the lineup whose win probability can be altered by the producers without changing the rules of the game?

Going down the list, I only came up with one more - Magic #.

You could make a case for games like Cliff Hangers, Spelling Bee, and Rat Race, I suppose - if you make the products, say, $300 each, nobody's going to win anything, but within reason, I don't think they count.

Grocery game could theoretically be manipulated in a similar way to Pay the Rent, in that the number of winning combinations changes based on the prices of the groceries picked. However, the number of possible combinations (winning or losing) is big enough that it's the kind of thing that would have to be programmed rather than done by hand or in a spreadsheet.

Super Saver and Buy or Sell can be manipulated by making it easier to come back from a wrong decision, since it's the cumulative total of your decisions that matters.

For Bullseye, there are prices (all in the sub-$2.00 range) that have more than one multiple between $10 and $12.