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, drops the prize by $250,000 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.
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.
Probabilities
Truth BoothTaylor and Nathan go into the Truth Booth at the end of Episode 1. The start of Episode 2 shows... no match.
| Aqel | Brendan | Clay | Eduardo | Hamudi | Leo | Mikey | Nathan | Ollie | Samuel | Will | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Anissa | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 10.0% | 9.0% | 9.0% | 9.0% |
| Brooke | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 10.0% | 9.0% | 9.0% | 9.0% |
| CC | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 10.0% | 9.0% | 9.0% | 9.0% |
| Courtney | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 10.0% | 9.0% | 9.0% | 9.0% |
| Danielle | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 10.0% | 9.0% | 9.0% | 9.0% |
| Dew | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 10.0% | 9.0% | 9.0% | 9.0% |
| Jordanne | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 10.0% | 9.0% | 9.0% | 9.0% |
| Julia Ruth | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 10.0% | 9.0% | 9.0% | 9.0% |
| Mijntje | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 10.0% | 9.0% | 9.0% | 9.0% |
| Roz | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 9.0% | 10.0% | 9.0% | 9.0% | 9.0% |
| Taylor | 10.0% | 10.0% | 10.0% | 10.0% | 10.0% | 10.0% | 10.0% | X | 10.0% | 10.0% | 10.0% |
36,288,000 possibilities remain.
Matchup Ceremony
- Nathan - Mijntje
- Leo - Brooke
- Mikey - Danielle
- Brendan - CC
- Aqel - Courtney
- Hamudi - Taylor
- Ollie - Anissa
- Samuel - Julia Ruth
- Eduardo - Roz
- Clay - Dew
- Will - Jordanne
Blackout probability: 36.4% (13,216,113 of 36,288,000)
Instant win possible: Yes
Most likely number of lights: 1
It's a blackout! 0 beams. 13,216,113 combinations left.
| Aqel | Brendan | Clay | Eduardo | Hamudi | Leo | Mikey | Nathan | Ollie | Samuel | Will | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Anissa | 9.9% | 9.9% | 9.9% | 9.9% | 10.0% | 9.9% | 9.9% | 11.1% | X | 9.9% | 9.9% |
| Brooke | 9.9% | 9.9% | 9.9% | 9.9% | 10.0% | X | 9.9% | 11.1% | 9.9% | 9.9% | 9.9% |
| CC | 9.9% | X | 9.9% | 9.9% | 10.0% | 9.9% | 9.9% | 11.1% | 9.9% | 9.9% | 9.9% |
| Courtney | X | 9.9% | 9.9% | 9.9% | 10.0% | 9.9% | 9.9% | 11.1% | 9.9% | 9.9% | 9.9% |
| Danielle | 9.9% | 9.9% | 9.9% | 9.9% | 10.0% | 9.9% | X | 11.1% | 9.9% | 9.9% | 9.9% |
| Dew | 9.9% | 9.9% | X | 9.9% | 10.0% | 9.9% | 9.9% | 11.1% | 9.9% | 9.9% | 9.9% |
| Jordanne | 9.9% | 9.9% | 9.9% | 9.9% | 10.0% | 9.9% | 9.9% | 11.1% | 9.9% | 9.9% | X |
| Julia Ruth | 9.9% | 9.9% | 9.9% | 9.9% | 10.0% | 9.9% | 9.9% | 11.1% | 9.9% | X | 9.9% |
| Mijntje | 10.0% | 10.0% | 10.0% | 10.0% | 10.1% | 10.0% | 10.0% | X | 10.0% | 10.0% | 10.0% |
| Roz | 9.9% | 9.9% | 9.9% | X | 10.0% | 9.9% | 9.9% | 11.1% | 9.9% | 9.9% | 9.9% |
| Taylor | 11.1% | 11.1% | 11.1% | 11.1% | X | 11.1% | 11.1% | X | 11.1% | 11.1% | 11.1% |
For an explanation on why Mijntje and Hamudi have slightly higher probabilities than everyone else (i.e. the difference between 9.9%, 10.0%, and 10.1%), please see the comments section.
Glad to see you back Alex!! <3
ReplyDeleteGlad to see the blog lives on! Can't believe they blacked out the first time, lol this show is great
ReplyDeleteIt's not so unbelievable; the chances of a blackout in the first ceremony are always about 1/3, so it's actually surprising it doesn't happen more. Of course, it's not up to random chance; in theory, they are making decisions about something knowable.
DeleteOmg you're back!!! 💜💜💜
ReplyDeleteWhere we see 9.9%, 10.0%, and 10.1%, is that a rounding error among values that should all be equal to each other? I would think that all remaining possible matches not involving Taylor and Nathan would have equal probability.
ReplyDeleteIt's not a rounding error - they are actually 9.8625%, 9.9888%, and 10.1010%. In absolute numbers: 1303441, 1319994, and 1334961. Mijntje and Hamudi are slightly higher to be a match themselves since they were in the matchup ceremonies with Nathan and Taylor.
DeleteThe explanation is something like this, not particularly intuitive:
1. Nathan and Taylor have the most information, they are equally likely to be with any of the contestants they didn't match with this week (1/9 = 11.1%).
2. Each of those 9 men and 9 women are slightly more likely to be with Taylor or Nathan than the 9 others who they have no information on, but are equally likely to be with any of those remaining 9 ((100% - 11.1%) / 9 = 9.9%).
3. Each time you pick partners for Nathan and Taylor, that leaves 9 men and 9 women, including Hamudi and Mijntje. Hamudi and Mijntje have 8 possible partners, but the other 8 men and 8 women only have 7 possible partners. Thus, you would expect that Hamudi and Mijntje are each slightly more likely to be the other 8's matches than everyone else (the explanation for 10.0% > 9.9%).
4. Now let's pick Hamudi's match third after Nathan and Taylor. If it is Mijntje, that leaves 8 women who each have 7 or 8 possible matches (depending on if Taylor is with their confirmed no-match). If Hamudi's partner is one of the 8 other women, then that leaves Mijntje with 8 possible matches while the remaining 7 women each have 7 or 8 possible matches. The logic to tally up the totals here is too complicated to explain in a single paragraph, but it ends up showing Hamudi with Mijntje more often than other women (the explanation for 10.1% > 10.0%).
Oh, okay, that makes sense. Thanks for taking the time to explain!
DeleteHappy to see you and this math back!
ReplyDeleteThe blackout in round one is surprising, not because it's statistically implausible, but because they've never let it happen before. In four of the eight seasons, the last couple to pair up at the first ceremony was a perfect match. This can't be a coincidence, it has to be producer interference to boost the matches. Since they have contestants prepare an ordered list of who they would like to choose (which they must stick to) it's easy for production to intentionally rig it so the two people leftover at the end are a beam.
So, most likely, one of two things happened: 1) With the move to streaming, Paramount is not trying to give away a million and decided not to assist in the way it normally does round one. 2) All the players agreed who they were picking before the ceremony and they didn't step on each other's toes for who they put down as their #1 choice. This is a good strategy for later in the game to ensure that you get the exact set of matches you want without production interfering by putting someone who goes rogue and triggers a domino effect with the other picks, but it is not useful in the first ceremony since production can do nothing to help you if you all agree to who gets who ahead of time.
Oh, that's really interesting. I've never been in the "Rigged!" camp, but I didn't know about the ordered lists. It certainly makes sense that the producers would manipulate that, both for drama and to influence the results.
ReplyDeleteYay! You're back!!
ReplyDeleteOMGGG So glad you are backed!! Wasn't sure if you were aware of the new season coming out but glad you're watching along with us and doing the math again >_<
ReplyDelete