This page describes in detail the math behind the odds of how Pay the Rent works, and how the show has conducted the game.
Possible CombinationsEdit
First consider that the shopping items are ranked in price from 1 (lowest) to 6 (highest). So a combination could be 351426, with the 3rd most expensive item in the Mailbox, and so on until the highest priced item (6) is in the Attic.
When selected, the 'Mailbox' and 'Attic' are single items, while the two middle 'Floors' are two items each. The total possible combinations of choices are therefore:
- Mailbox: 6 possibilities
- 1st Floor: 10 possibilities (in math, this is: 5 Items, Choose 2)
- 2nd Floor: 3 possibilities (3 Items, Choose 2)
- Attic: 1 possibility
So 6 x 10 x 3 x 1 = 180 total combinations.
Since the only way to win is to put the highest priced item in the Attic, this must be selected first; which changes the odds:
- Mailbox: 5 possibilities (because the Attic has already been chosen)
- 1st Floor: 6 possibilities (4 Items, Choose 2)
- 2nd Floor: 1 possibility (2 Items, Choose 2)
- Attic: 1 possibility
So 5 x 6 x 1 x 1 = 30 total combinations.
The 30 possible combinations that could win are:
123456 | 234516 | 345126 | 451236 | 512346 | ||||
124356 | 235416 | 341526 | 452136 | 513246 | ||||
125346 | 231456 | 342516 | 453126 | 514236 | ||||
134256 | 245316 | 351426 | 412536 | 523146 | ||||
135246 | 241356 | 352416 | 413526 | 524136 | ||||
145236 | 251346 | 312456 | 423516 | 534126 |
Odds Edit
While there appears to be a 1 in 30 chance of choosing the right combination (after correctly selecting the highest-priced item), the actual odds are much different since they're driven by the price-structure chosen by the show. This allows the show some control over the frequency of wins, though there's always a chance to win.
The percent increase between the two highest priced items is the primary driver in determining the number of successful combinations. If the #5 and #6 prices are close, there will likely be only one or two successful combinations. A large difference in the price of these two items will greatly increase the number of successful combinations, and odds of someone winning. To a lesser extent, the price difference between the middle items can make the difference between having one to four successful combinations, but the value of the low-priced item (and difference between it and the 2nd item) won't matter much.
If the price increase from item-to-item is low (based on the percent increase), there is likely to be only one possible winning combination where the price will increase from the Mailbox to the Attic. For this case, there will typically only be a 10% to 20% increase between the 2nd through 6th item. A good example is the very first time the game was played. The prices were: $1.49, $2.98, $3.49, $5.49, $5.99, and $7.30. The percent increase between each was: 100%, 17%, 57%, 9%, and 22%. Again, the first increase doesn't matter much, but the small increase between the 5th and 6th item, along with low increases in the others made this game have only one successful combination (523146).
The first 27 games obeyed this price structure fairly closely. Game 28 started a new trend were the percent increase was altered. A good example is Game 29 where the percent increases were: 85%, 110%, 48%, 71%, and 73%. This was the first large difference between the 5th and 6th item, and allowed for 8 successful combinations.
The greatest disparity was in a game with 14 successful combinations that resulted in the contestant winning. In this game, the percentage increases between prices was: 10%, 68%, 11%, 22%, and 300%. Tripling the price between the 5th and 6th item caused the high number of successful combinations, making it nearly a 50/50 coin-toss that the contestant would win.
Frequency of Successful Combinations Edit
As of 3 Jun 2016, the game has been played 56 times, resulting in 150 combinations that could have won. If the shopping items were chosen at random, the frequency of successful combinations would be about the same.
This is not the case, presumably due to the show using a standard formula for selecting items based on the previous section's discussion. In fact, of the 30 possible successful combinations, 19 have been used, and only 6 combinations with any frequency.
Though, knowing only 2 combinations would have won 30 of 56 games. The chart below shows the number of times a combination would have been successful based on the number of winning combinations in a game. For instance, there have been 4 games played with 8 winning combinations; and combination 124356 was in all of them.
Possible winning combinations in Game | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|
Winning sequence |
1 |
2 |
3 |
4 |
6 |
7 |
8 |
10 |
14 |
Total |
123456 |
2 |
1 |
1 |
4 | ||||||
124356 | 1 | 1 | 4 | 1 | 1 | 8 | ||||
125346 |
1 |
3 |
1 |
5 | ||||||
134256 |
1 |
1 |
1 |
1 |
4 | |||||
231456 |
2 |
1 |
1 |
4 | ||||||
234516 |
2 |
1 |
1 |
4 | ||||||
241356 |
1 |
1 |
4 |
1 |
1 |
8 | ||||
251346 |
5 |
1 |
1 |
3 |
1 |
1 |
12 | |||
312456 |
1 |
1 |
2 | |||||||
341526 |
2 |
1 |
1 |
4 |
1 |
1 |
10 | |||
342516 |
2 |
3 |
3 |
2 |
1 |
1 |
2 |
1 |
15 | |
351426 |
2 |
3 |
5 |
2 |
1 |
13 | ||||
412536 |
1 |
1 | ||||||||
413526 |
1 |
1 |
1 |
3 | ||||||
423516 |
8 |
8 |
6 |
1 |
1 |
1 |
1 |
1 |
27 | |
451236 |
4 |
2 |
2 |
1 |
9 | |||||
513246 |
1 |
1 |
2 | |||||||
514236 |
5 |
2 |
1 |
8 | ||||||
523146 |
6 |
3 |
1 |
1 |
11 | |||||
Total combinations |
27 |
22 |
24 |
8 |
6 |
7 |
32 |
10 |
14 |
150 |
# Games Played | 27 | 11 | 8 | 2 | 1 | 1 | 4 | 1 | 1 | 56 |
So the combination 4 - 23 - 51 - 6 would have won over half of all the games played.
Thus far the show appears to have only used two basic approaches for deciding the price structure. One will result in 1 to 3 possible solutions, the other will have 6 or more solutions.
In the first price structure (46 games), where prices are close together, the most successful combinations and their frequency of occurrence were:
342516 | - | 8 |
351426 | - | 10 |
423516 | - | 22 |
---|---|---|
451236 | - | 8 |
514236 | - | 7 |
523146 | - | 10 |
So in the 46 games, only combination 4 - 23 - 51 - 6 has a good probability of winning.
In the second price structure (8 games with 6 or more winning combinations) where the highest priced item is much more expensive than the 5th item, there are many more successful combinations, but only 3 occur in every game:
% occurrence | |||
124356 | - | 8 | 100% |
---|---|---|---|
241356 | - | 8 | 100% |
341526 | - | 8 | 100% |
The combination 4 - 23 - 51 - 6 also occurs in half of these games.
Therefore, always using combination 4 - 23 - 51 - 6 (or using combination 1 - 24 - 35 -6 if the the highest priced item is much more expensive) shows the best opportunity to win.
How to Win! Edit
The preceding relies heavily on being able to discern the item prices and the price structure. Additionally, the game is played in reverse from the way it should actually be thought through (starting at the Mailbox rather than the Attic).
A better approach 'in the heat of the moment' is to put two items each into three categories (Low, Medium, and High). In this case, generally the best odds are to put a (Medium) in the Mailbox, a (Low) and (Medium) in the 1st floor, a (Medium) and (High) in the 2nd floor, and a (High) in the Attic.
As of 3 March, 2016, this strategy accounts for over 40% of the 138 successful combination to date and over 50% over the high-odds games.
Unfortunately, if the price structure is for a low-odds game, the odds of winning remain extremely low; primarily driven by the difficulty in selecting the high-priced item, and the courage required to 'pass' on $10,000 when the Attic item may be only 30-cents more than the 2nd Floor total.