So, after a week, I return to the math again.
If I calculated correctly, the answer to all three of my questions is
the same. The number of pots you expect to have to smash (again assuming your opponents don't give anything away) is:
1)
5.5 pots if you're looking for 1 hidden coin (what Green faced in Episode 2, and Yellow faced in Episode 3)
2)
5.5 pots if you need 2 coins and your opponents hid 3 (what Blue faced in Episode 6, and what Pink faced in Episode 7)
3)
5.5 pots if you need 3 coins and your opponents hid 5 (which did not happen during the game, but could have happened in Episode 4 if the coins given to the Gray and Yellow teams were split 3-1 instead of 2-2)
Clearly, a lot of the "poker techniques" the contestants used in the Cave of Karma worked, because of all 6 eliminations at the Cave of Karma, only the elimination in Episode 4 (Gray eliminating Yellow) resulted in more than the expected number of pots being smashed.
Here are the average expected pots smashed (with no information given away - that is, random guessing) for all configurations that came up in the game, or potentially could have come up based on how the Karma coins were distributed:
To find 1 coin:
Smash
5.5 pots if only 1 coin was hidden (Green in Episode 2, Yellow in Episode 3)
Smash
3.67 pots if 2 coins were hidden (never occurred, but this is included for comparison; this could have happened if they didn't increase the number of starting coins given to each team from 1 to 2 starting in Episode 4)
Smash
2.75 pots if 3 coins were hidden (never occurred, but this is included for comparison)
Smash
2.2 pots if 4 coins were hidden (never occurred, but this is included for comparison)
Smash
1.83 pots if 5 coins were hidden (Orange in Episode 3)
Smash
1.57 pots if 6 coins were hidden (Yellow in Episode 2)
Smash
1.38 pots if 7 coins were hidden (never occurred, but could have if they awarded a bonus coin for the team that won the first challenge like they did after Tilt, AND that team used their bonus coin on the team that already had the most coins, as it was for Yellow in Episode 2)
To find 2 coins:
Smash
7.33 pots if only 2 coins were hidden (Red in Episode 6, Purple in Episode 7)
Smash
5.5 pots if 3 coins were hidden (Blue in Episode 6, Pink in Episode 7)
Smash
4.4 pots if 4 coins were hidden (never occurred, but could have if Purple gave their coin to Blue instead of Gray in Episode 5)
Smash
3.67 pots if 5 coins were hidden (never occurred, but could have if Red gave their coin to Green instead of Yellow in Episode 2; Yellow would have been looking for 2 coins from 5 hidden)
Smash
3.14 pots if 6 coins were hidden (never occurred, but could have if Blue gave their coin and their bonus coin to Yellow instead of Gray in Episode 4)
To find 3 coins:
Smash
8.25 pots if only 3 coins were hidden (Gray in Episode 5, Blue in Episode 5)
Smash
6.6 pots if 4 coins were hidden (never occurred, but would have happened for Yellow in Episode 4 if Blue had not given Gray their bonus coin for winning Tilt)
Smash
5.5 pots if 5 coins were hidden (never occurred, but would have if the coins given out had been split 3-1 between Yellow and Gray in Episode 4)
To find 4 coins:
Smash
8.8 pots if only 4 coins were hidden (Yellow in Episode 4, Gray in Episode 4)
---
A more interesting question than how many pots you would expect to smash is what percent of the time you would expect to win based on how many coins you hide for your opponents and how many coins your opponents hide for you. This requires knowledge of how the producers decided to break ties. Based mostly on what was discussed in the RHAP Episode 2 recap, where it was revealed that Green Team actually hit a coin on their first smash, but then also Yellow hit a coin on their first smash, this is my understanding:
*If a team gets the number of coins they need to win with the minimum number of smashes (that is, 1 smash in both Episode 2 and Episode 3), the other team gets a chance to try to match them. If the other team also finds the number of coins they need to win with the minimum number of smashes, then it's considered a tie and they reset the whole setup. (So that meant that once Yellow got their coin after Green got their coin in Episode 2, they put out new pots and had Yellow hide 1 coin for Green, and Green hide 6 coins for Yellow. What was shown in Episode 2 was what happened after the reset.)
*Once a team misses getting a coin with a smash, the first team to find the number of coins needed automatically wins. So the teams are NOT guaranteed to get to smash the same number of pots. I saw in the Episode 6 thread that there was a question about why Blue did not get a chance to equalize after Red found their second coin; this is why - both teams had had a miss before Red found their second coin. The upshot of this is that going first is a really important advantage.
So again, starting with assumption that no extra information is given away (both teams have good poker faces or turn around), here are the percentages I calculated:
To find 1 coin:
*If the team with more coins had 7 coins hidden, there's a 7% chance of a tie and reset, a 6.75% chance that the team with only 1 coin wins without a tie occurring, and a 86.25% chance that the team with 7 coins wins without a tie occurring. Normalizing to remove the chance of a tie, the team with only 1 coin wins 7.26% of the time and the team with 7 coins wins 92.74% of the time.
*If the team with more coins had 6 coins hidden, there's a 6% chance of a tie and reset, a 9.71% chance that the team with only 1 coin wins without a tie occurring, and a 84.29% chance that the team with 6 coins wins without a tie occurring. Normalizing to remove the chance of a tie, the team with only 1 coin wins 10.33% of the time and the team with 7 coins wins 89.67% of the time.
*If the team with more coins had 5 coins hidden, there's a 5% chance of a tie and reset, a 13.33% chance that the team with only 1 coin wins without a tie occurring, and a 81.67% chance that the team with 6 coins wins without a tie occurring. Normalizing to remove the chance of a tie, the team with only 1 coin wins 14.04% of the time and the team with 7 coins wins 85.96% of the time.
To find 2 coins:
*If the team with more coins had 6 coins hidden, there's a 0.74% chance of a tie and reset, a 7.99% chance that the team with only 2 coins wins without a tie occurring, and a 91.27% chance that the team with 6 coins wins without a tie occurring. Normalizing to remove the chance of a tie, the team with only 2 coins wins 8.05% of the time and the team with 6 coins wins 91.95% of the time.
*If the team with more coins had 5 coins hidden, there's a 0.49% chance of a tie and reset, a 12.31% chance that the team with only 2 coins wins without a tie occurring, and a 87.20% chance that the team with 5 coins wins without a tie occurring. Normalizing to remove the chance of a tie, the team with only 2 coins wins 12.37% of the time and the team with 5 coins wins 87.63% of the time.
*If the team with more coins had 4 coins hidden, there's a 0.30% chance of a tie and reset, a 19.26% chance that the team with only 2 coins wins without a tie occurring, and a 80.44% chance that the team with 4 coins wins without a tie occurring. Normalizing to remove the chance of a tie, the team with only 2 coins wins 19.32% of the time and the team with 4 coins wins 80.68% of the time.
*If the team with more coins had 3 coins hidden, there's a 0.15% chance of a tie and reset, a 31.63% chance that the team with only 2 coins wins without a tie occurring, and a 68.22% chance that the team with 3 coins wins without a tie occurring. Normalizing to remove the chance of a tie, the team with only 2 coins wins 31.68% of the time and the team with 3 coins wins 68.32% of the time.
To find 3 coins:
*If the team with more coins had 5 coins hidden, there's a 0.069% chance of a tie and reset, a 15.64% chance that the team with only 3 coins wins without a tie occurring, and a 84.29% chance that the team with 5 coins wins without a tie occurring. Normalizing to remove the chance of a tie, the team with only 3 coins wins 15.66% of the time and the team with 5 coins wins 84.34% of the time.
*If the team with more coins had 4 coins hidden, there's a 0.028% chance of a tie and reset, a 29.57% chance that the team with only 3 coins wins without a tie occurring, and a 70.40% chance that the team with 4 coins wins without a tie occurring. Normalizing to remove the chance of a tie, the team with only 3 coins wins 29.58% of the time and the team with 4 coins wins 70.42% of the time.
*If both teams had 3 coins hidden, there's a 0.0069% chance of a tie and reset, a 60.03% chance that the team that goes first wins without a tie occurring, and a 39.96% chance that the team that goes second wins without a tie occurring. Normalizing to remove the chance of a tie, the team that goes first wins 60.04% of the time and the team that goes second wins 39.96% of the time.
To find 4 coins:
*If both teams have 4 coins hidden, there's a 0.0027% chance of a tie and reset, a 63.53% chance that the team that goes first wins without a tie occurring, and a 36.47% chance that the team that goes second wins without a tie occurring. Normalizing to remove the chance of a tie, the team that goes first wins 63.53% of the time and the team that goes second wins 36.47% of the time (normalization has a very tiny decimal effect because the chance of a tie and reset is so low).
In table form with all the normalization performed:
Leading Team Coins
| Leading Team Wins
| Trailing Team Coins
| Trailing Team Wins
|
7
| 92.74%
| 1
| 7.26%
|
6
| 89.67%
| 1
| 10.33%
|
5
| 85.96%
| 1
| 14.04%
|
6
| 91.95%
| 2
| 8.05%
|
5
| 87.63%
| 2
| 12.37%
|
4
| 80.68%
| 2
| 19.32%
|
3
| 68.32%
| 2
| 31.68%
|
5
| 84.34%
| 3
| 15.66%
|
4
| 70.42%
| 3
| 29.58%
|
3*
| 60.04%
| 3
| 39.96%
|
4*
| 63.53%
| 4
| 36.47%
|
*Leading team leads because they get to smash first.
Takeaways:
*As more coins need to be found, the chances of a tie and reset decrease quickly. This is good from the perspective of the producers, who probably don't like to have to reset and re-shoot.
*As expected, the most impressive instance of a team beating the odds in Karma season 1 was Yellow Team beating Orange Team in Episode 3.
*If Red Team had tried to help Green Team in Episode 2 by not falling in line with the other teams and giving Green Team their coin, they only would have changed the odds that Green Team would win by a few percentage points (10.33% chance to win at 1-6 down versus 12.37% chance to win at 2-5 down).
*Blue Team giving up their bonus coin for winning Tilt to Gray Team in Episode 4 changed Gray Team's odds from 29.58% (at 3-4) to 63.53% (at 4-4 smashing first).
*Purple Team's attempt to secure Gray Team's medallions by giving them a coin decreased Blue Team's odds from 80.68% (at what would have been 4-2) to 60.04% (at 3-3 smashing first). That's a significant decrease; good thing for Purple Team that the mumbo-jumbo magic worked.
*I would like to know how they decided which team got to smash first when the number of coins each team had was even. In Episode 4, with Gray Team and Yellow Team having 4 coins each, Gray Team smashed first and won. In Episode 5, with Blue Team and Gray Team having 3 coins each, Blue Team smashed first and won. Going first is important!
*The difference between being 3-2 up in coins (but thereby having the opponent go first) and being even at 3-3 in coins but getting to smash first seems surprisingly small (68.32% to win at 3-2 versus 60.04% to win at 3-3 smashing first, less than 9% difference).
With all that said, I am not sure I have all the math correct. Help, corrections, comments, critiques, and suggestions are all welcome.
