Here I'll be keeping track of the possible remaining perfect match combinations (with the help of a computer), and looking into the potential strategy of playing this game. The math work here will be independent of the romance and excitement that comes with the pairings. Unless noted otherwise, assume the probabilities of a matchup here being successful are random, ignoring chemistry and all the parts of the show that are actually fun.
Prize and Probability of Winning
All 22 winners will split a 30-year annuity that will pay out $1,000,000. That means if they elect to take the present day "lump sum," they'll end up with around $22,000 each (in addition to a shot at finding their perfect match and getting some weekly face time on MTV).The blackout rule, introduced in season 3, cuts the prize in half if a matchup ceremony results in no lights other than truth booths.
The number of possible permutations in the original game is 11 factorial = 11! = 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 39,916,800. Put simply, a random guess would have a 1 in 39,916,800 chance of winning (0.00000002505%). The calculations done here throughout the season evaluate all of those 39,916,800 possibilities against the information that's been shared through truth booths and matchup ceremonies to figure out the set of perfect matches. The easy way of saying it: I use a computer program to do a process of elimination in seconds.
Previous seasons have used 20 contestants - by adding another couple to the house, MTV has made this season 11 times harder than the four previous seasons.
The show has multiple challenges where additional information is provided, reducing the number of possible combinations. The contestants only have 10 episodes to find the perfect match, so they'll need to reduce it quickly to have a chance.
Live Blog
9:00: Welcome back for Season 5! As mentioned above, this season now has 22 contestants, and getting all of the perfect matches is now 11 times harder. Is MTV setting us up for the first failure, or will they rig it again?9:01: They've moved from Hawaii to the Dominican Republic!
9:03: The parents are playing matchmaker, selecting three dates for the getaway date: Joey - Carolina, Hannah - Osvaldo, and Gianna - Hayden.
9:04: The parents are setting up couples where the girl is not in to the guy at all!
9:06: Casandra can count!
9:07: Haircut, Eyebrows, And a Tan.
9:08: A girl who loves animals??
9:09: Also, Hayden, her mom picked you!
9:10: Spreading seed to see what bites?
9:11: Michael with a good line: "I dated a lot of girls with holes [in their hearts] but none of them got the surgery."
9:13: Kathryn and Mike going for it - in front of everyone!!!
9:18: Might not be the indecision that's making your stomach hurt...
9:20: Hayden freaking out Gianna. Too weird for her!
9:21: The AfterMatch is back!
9:26: The getaway is riding a truck through a forest and river.
9:28: "I would never cheat on a girl." "I like that about you." (Cue happy music and a kiss!)
Truth Booth
9:29: Hayden and Gianna to the Truth Booth!
9:30: Twist!! "What is the truth booth worth to you?"
9:34: Ryan offers the house a deal - give up the truth booth and the right to ever send that couple to the truth booth, get an extra $150,000 added to the prize.
9:35: Derrick knows what's up - the first episode, where they know nothing (9.1%), is probably the best time to add this money. And yet they choose not to take the trade!
9:36: No match means 36,288,000 combinations remain.
| Andre | Derrick | Edward | Hayden | Jaylan | Joey | Michael | Mike | Osvaldo | Ozzy | Tyler | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Alicia | 9.0% | 9.0% | 9.0% | 10.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% |
| Carolina | 9.0% | 9.0% | 9.0% | 10.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% |
| Casandra | 9.0% | 9.0% | 9.0% | 10.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% |
| Gianna | 10.0% | 10.0% | 10.0% | X | 10.0% | 10.0% | 10.0% | 10.0% | 10.0% | 10.0% | 10.0% |
| Hannah | 9.0% | 9.0% | 9.0% | 10.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% |
| Kam | 9.0% | 9.0% | 9.0% | 10.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% |
| Kari | 9.0% | 9.0% | 9.0% | 10.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% |
| Kathryn | 9.0% | 9.0% | 9.0% | 10.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% |
| Shannon | 9.0% | 9.0% | 9.0% | 10.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% |
| Taylor | 9.0% | 9.0% | 9.0% | 10.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% |
| Tyranny | 9.0% | 9.0% | 9.0% | 10.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% |
Some very loose estimates on expected values on the truth booths... the estimate here is based on the expected number of combinations produced by the truth booth, multiplied by the prize and number of matchup ceremonies remaining. It does not account for the value of future truth booths, blackouts, or that each subsequent matchup ceremony reduces the number of combinations remaining - all of which could significantly impact the calculation!
Estimated value of forfeiting truth booth for $150,000: $1.53
Estimated value of going to the truth booth: $1.61
Estimated value of losing the truth booth: $1.47
It's worth trading the truth booth for approximately: $198,020
Matchup Ceremony
The new blackout rule is the prize gets cut in half - not just losing $250,000!
- Kam - Edward
- Taylor - Tyler
- Kari - Mike
- Casandra - Jaylan
- Carolina - Joey
- Tyranny - Osvaldo
- Gianna - Ozzy
- Hannah - Michael
- Alicia - Andre
- Kathryn - Derrick
- Shannon - Hayden
Instant win possible: Yes
Most likely number of lights: 1
Two matches mean 6,741,548 possibilities remain.
| Andre | Derrick | Edward | Hayden | Jaylan | Joey | Michael | Mike | Osvaldo | Ozzy | Tyler | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Alicia | 17.8% | 8.2% | 8.2% | 8.9% | 8.2% | 8.2% | 8.2% | 8.2% | 8.2% | 8.0% | 8.2% |
| Carolina | 8.2% | 8.2% | 8.2% | 8.9% | 8.2% | 17.8% | 8.2% | 8.2% | 8.2% | 8.0% | 8.2% |
| Casandra | 8.2% | 8.2% | 8.2% | 8.9% | 17.8% | 8.2% | 8.2% | 8.2% | 8.2% | 8.0% | 8.2% |
| Gianna | 8.9% | 8.9% | 8.9% | X | 8.9% | 8.9% | 8.9% | 8.9% | 8.9% | 19.8% | 8.9% |
| Hannah | 8.2% | 8.2% | 8.2% | 8.9% | 8.2% | 8.2% | 17.8% | 8.2% | 8.2% | 8.0% | 8.2% |
| Kam | 8.2% | 8.2% | 17.8% | 8.9% | 8.2% | 8.2% | 8.2% | 8.2% | 8.2% | 8.0% | 8.2% |
| Kari | 8.2% | 8.2% | 8.2% | 8.9% | 8.2% | 8.2% | 8.2% | 17.8% | 8.2% | 8.0% | 8.2% |
| Kathryn | 8.2% | 17.8% | 8.2% | 8.9% | 8.2% | 8.2% | 8.2% | 8.2% | 8.2% | 8.0% | 8.2% |
| Shannon | 8.0% | 8.0% | 8.0% | 19.8% | 8.0% | 8.0% | 8.0% | 8.0% | 8.0% | 7.9% | 8.0% |
| Taylor | 8.2% | 8.2% | 8.2% | 8.9% | 8.2% | 8.2% | 8.2% | 8.2% | 8.2% | 8.0% | 17.8% |
| Tyranny | 8.2% | 8.2% | 8.2% | 8.9% | 8.2% | 8.2% | 8.2% | 8.2% | 17.8% | 8.0% | 8.2% |
Not sure the average viewer understands how much more difficult this season is. With these rules, production must really want them to lose. (Makes sense though, the game looks too "easy" after four straight wins.)
ReplyDeleteSo happy you're back! I like that they made the season more difficult to win, but I also feel like it'll make it much more clear if they rig it again. I like that you broke down the math on the value of the truth booth. I assumed they should always go with gaining information on matches, but your argument made a lot of sense.
ReplyDeleteMy loose estimate of the numbers actually disagree with my suggestion in the live blog. This truth booth was probably worth around $200,000 in increased prize money - so it wasn't worth it to trade.
DeleteI don't understand your claim that the show is rigged. Competition shows like this one have standards and practices requirements designed to stop the sort of rigging you describe.
ReplyDeleteThe only reason I suggest this is that during season 3, the house went into the final matchup ceremony without the set of perfect matches being revealed. That is - if they went through every one of the 8! = 40,320 combinations and checked them (an extreme difficulty given that they were not even allowed pen and paper), they would have still been left over with 4 different possibilities. So even if you have the faith in this house to reduce those combinations correctly, that still only got them to a 1 in 4 chance of getting it right.
DeleteAdditionally, I can totally understand standards and practices that would protect the contestants from this show, or people competing against each other - but what about a scenario where the production company giving away the money did something where the contestants were more likely to win the money? That doesn't seem as problematic.
Finally, I'll just leave this here: https://www.reddit.com/r/BravoRealHousewives/comments/4ma3vs/ive_worked_on_several_reality_and_scripted/d3twark/
Thanks for the link.
Delete"Rigged to win" is very far away from a PA accidentally slipping a hint about a perfect match, though I'm surprised they would even have access to that information. Even one such hint could change the 1 in 4 to a 100% hit. (And things that are 1 in 4 do happen ... about 25% of the time!)
If the show were truly rigged, there is no way the contestants would have been so far away going into the final episode. And while it may have happened, a hint from a PA should be against S&P. I'll bet if that did happen they would be very careful going forward not to do that again.
Based on comments you made during Season 4, you seemed convinced the show is rigged based on how that season went as well. If the contestants won in Seasons 1, 2, and 4 only, would you still feel it was rigged?
Thanks for the site. It'll be interesting to see how the S5 changes (11 pairs, opportunity to skip Truth Booth) impact the contestants' chances of winning.
Rigged is probably strong wording, but I do believe that the outcome of season 3 was unfairly influenced. I raised my eyes a bit at season 4, but the extremely low probability of success in season 3 is the main source of my skepticism.
DeleteWhile I still believe it would be extremely hard to narrow down those 39 million possibilities without pen and paper, on further reflection, I actually think the team could divide the responsibilities of remembering the rules of each matchup ceremony and truth booth among the contestants and it could be a matter of everyone figuring it out in a room together for a couple hours.
Even after taking a look at that link I find it very hard to believe that season 3 was rigged. Even if they were provided a "hint", I'm doubtful it got them anywhere closer than the 25% chance they had. Really think that one just boils down to dumb luck. Whereas season 4 mathwise had it in the bag for awhile so it's only a matter of time before someone puts that together. No offense Alex, love your blog, I just love reality tv competition shows and really hate when people decide to cry wolf ("rigged")
DeleteThe variables! As the show is not "love or money" but, rather "if love, then money" (provided we agree that "love" is what the matchmakers have calculated, not production value), it would not make sense to eschew a truth booth for additional funds (without additional information such that the truth booth were rendered superfluous or enough chances remained that the couple could be determined through the standard lighting ceremony) as the money cannot be procured without a win.
ReplyDeleteI am very excited to see how this shakes out strategically, particularly since money in exchange for truth booth adds a new expected value calculation to the equation. I have a feeling that they added the extra pairing to ensure they received a full order of episodes with more emotional turmoil owing to the addition of an 11th pairing. (11!)
The cast itself seems lackluster compared to the statistics at this point - more so than other seasons. It makes the math less fun when the variables (aka cast members) seem like straightforward reality TV cutouts.
Awesome! So glad you are back tracking Season 5.
ReplyDelete%90 sure Kam and Edward are one of those two matches. In the previews they are the only two not celebrating when the truthbooth results are announced
ReplyDeleteCould you explain the math behind the probability table its killing me!
ReplyDeleteI just take all 39,916,800 combinations and remove those that don't fit what we've seen on the show. The probability is the percentage of combinations remaining that include that pairing. My computer program can check all of those combinations in seconds.
DeleteAh see im trying to write a computer program to do it haha thats why i asked!
DeleteHi Alex,
ReplyDeleteI saw your website some months ago and it made me want to code my own program. I did it in python and it works for the version with 10 pairings and the Season 2 version (I did some modifications there of course). But with 11 pairings, the function (itertools.permutations) I use to generate all the combinations takes infinite time. I was wondering how you did it ? Did you cut the data ? Or use a faster language ?
My program is in Java, which was my most comfortable language back when the show started. Memory is a concern in Java, as too little memory could result in an OutOfMemoryError (crash) or excessive garbage collection (slowdown). I currently use a byte to represent a pairing - if you were optimizing for memory usage, you could use a byte per matchup instead - which means storing the 39,916,800 combinations requires at least 420 MB of memory.
DeleteI don't know much about itertools or how Python handles memory management, but given the description of your problem, bumping up your memory allocation could solve the problem for you.
Hi Alex and Unknown, I've recently been working on my own program to calculate these same probability tables, and I did mine in Python.
DeleteI put the source up on github here: https://github.com/daturkel/ayto-calculator
It flies pretty nicely for 10x10, but 11x11 sure slows it down a lot.
That being said, I did use itertools.permutations to generate all of the combinations. Take a look here: https://github.com/daturkel/ayto-calculator/blob/master/rut1.py
I'd be happy to take a look at your code, if you'd like. And Alex, I'd be curious to hear what you think of mine (to the extent that you can make sense of it). Hoping to implement some of the other stats like chance of black out, most likely number of beams, but looking to create some better visualizations before then.
Alex,
ReplyDeleteCurious about your thoughts on the difficulty of this season's twist (11 men and 11 women) compared to season 2's. Just based on your permutations at the beginning of each blog it would seem that this season is more difficult, but isn't there an extra added layer of difficulty in season 2 that doesn't factor into the math with someone being left out in each match up ceremony is season 2 (it is the only season where you could lose on 9 correct matchups).
Also curious if you listen to the AYTO podcast on reality tv rhap ups (robhasapodcast)? I introduced them to your blog in season 2 and it pretty regularly gets mentioned! It'd be cool if you did a guest appearance.
http://robhasawebsite.com/?s=are+you+the+one
That's a good point, I haven't thought about the math at all!
DeleteI became aware of RHAP late last season - thanks for making that intro, it's really fun to hear someone talking about the blog. I haven't been invited on the show but that would be very exciting!
How are you calculating the chance of a blackout? How do you know at this stage if a couple would not be considered viable?
ReplyDeleteAlex's program works by taking all possible combinations of guy/girl and eliminating those that the truth booth/match up ceremonies have shown aren't possible. No matter what the actual correct matches are, mathematically a certain number of random combos will be the 11 perfect matches (only 1 random set of matchups), a certain number random will have 9 beams, 4 beams, 2 beams, 0 beam, etc. He just calculates what the percentage of 0 beam combinations remaining is.
DeleteFor an example of how this works, let's look at a much smaller group. Say the girls are A, B, and C, and the guys are 1, 2, 3. The correct combo is A-1, B-2, C-3 (it doesn't matter if the correct combo is actually something different, mathematically the probabilities are the same no matter what the labels are).
There are 6 possible ways to randomly combine the contestants:
A-1, B-2, C-3 (3 beams)
A-1, B-3, C-2 (1 beam)
A-2, B-1, C-3 (1 beam)
A-2, B-3, C-1 (0 beams)
A-3, B-1, C-2 (0 beams)
A-3, B-2, C-1 (1 beam)
2 of the 6 possible combos are 0 beams, or blackouts, therefore the chance of a random match-up being a black out is 2/6 or 33.33%.
But, say the group sends in a truth booth of matching A with 2 and finds out they are not a perfect match--then the 2 of the 6 combos containing A-2 can be eliminated (assuming girl A and guy 2 do not sit together at the matchup ceremony). The remaining combos are:
A-1, B-2, C-3 (3 beams)
A-1, B-3, C-2 (1 beam)
A-3, B-1, C-2 (0 beams)
A-3, B-2, C-1 (1 beam)
The odds of blackout are reduced to 1/4, or 25%.
Thanks for the detail!
DeleteI had worked it out this morning coming from a different angle.
I was thinking you are looking for all possible combinations that include none of the selected pairs divded by all of the possible combinations where there are at least one of the paired couples included. Is that the same?
Labelo - thanks for the great explanation!
DeleteTBB - I divide all of the remaining combinations that include none of the selected pairs by all of the remaining combinations. So, my equation is a/(a+b) whereas what you suggested in the previous comment was a/b.
Longtime reader!
ReplyDeleteI'm really curious to see if there will be an opportunity to add money to the pot by opting out of the truth booth every week. I suppose they wanted a way to increase the pot again since the cost of a blackout is so much higher this season, but doesn't MTV also depend on the inevitable breakups (or refusal to break up) caused when a lovestruck couple learns they aren't a match for much of of their drama? Taking away some of those moments seems a bad production move. Maybe they'll sometimes have challenges or other tasks worth money?
Then again, maybe it was a brilliant production move to add another element for the cast to fight and cry over. It's incredible to me that after four seasons some contestants still don't understand how valuable a no match is in the overall strategy of the game.
How are you calculating the probability of each match, and how does the beam count and truth booths affect it? Thanks!!
ReplyDelete