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 20 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 $25,000 each (in addition to a shot at finding their perfect match and getting some weekly face time on MTV).The number of possible permutations is 10 factorial = 10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 3,628,800. Put simply, a random guess would have a 1 in 3,628,800 chance of winning (0.00002756%).
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.
Truth Booth
No optimal strategy here. Just pick a couple and go into the Truth Booth, which would be a pretty boring room if you took out the TV and green lasers.Learning that a couple was a match would have eliminated 9 * 9! = 3,265,920 combinations, but learning that Shanley and Chris T weren't a match eliminated 9! = 362,880 possible combinations.
| Adam | Chris S | Chris T | Dillan | Dre | Ethan | Joey | John | Ryan | Wesley | |
|---|---|---|---|---|---|---|---|---|---|---|
| Amber | 9.9% | 9.9% | 11.1% | 9.9% | 9.9% | 9.9% | 9.9% | 9.9% | 9.9% | 9.9% |
| Ashleigh | 9.9% | 9.9% | 11.1% | 9.9% | 9.9% | 9.9% | 9.9% | 9.9% | 9.9% | 9.9% |
| Brittany | 9.9% | 9.9% | 11.1% | 9.9% | 9.9% | 9.9% | 9.9% | 9.9% | 9.9% | 9.9% |
| Coleysia | 9.9% | 9.9% | 11.1% | 9.9% | 9.9% | 9.9% | 9.9% | 9.9% | 9.9% | 9.9% |
| Jacy | 9.9% | 9.9% | 11.1% | 9.9% | 9.9% | 9.9% | 9.9% | 9.9% | 9.9% | 9.9% |
| Jessica | 9.9% | 9.9% | 11.1% | 9.9% | 9.9% | 9.9% | 9.9% | 9.9% | 9.9% | 9.9% |
| Kayla | 9.9% | 9.9% | 11.1% | 9.9% | 9.9% | 9.9% | 9.9% | 9.9% | 9.9% | 9.9% |
| Paige | 9.9% | 9.9% | 11.1% | 9.9% | 9.9% | 9.9% | 9.9% | 9.9% | 9.9% | 9.9% |
| Shanley | 11.1% | 11.1% | X | 11.1% | 11.1% | 11.1% | 11.1% | 11.1% | 11.1% | 11.1% |
| Simone | 9.9% | 9.9% | 11.1% | 9.9% | 9.9% | 9.9% | 9.9% | 9.9% | 9.9% | 9.9% |
3,265,920 combinations remain.
Matchup Ceremony
Obviously, the only restriction here is don't put Shanley and Chris T together.- Wesley - Kayla
- Ethan - Shanley
- Adam - Brittany
- Dre - Jacy
- John - Simone
- Chris T - Jessica
- Joey - Paige
- Chris S - Ashleigh
- Ryan - Amber
- Dillan - Coleysia
Out of those possible combinations, we have the following probabilities:
| Adam | Chris S | Chris T | Dillan | Dre | Ethan | Joey | John | Ryan | Wesley | |
|---|---|---|---|---|---|---|---|---|---|---|
| Amber | 8.9% | 8.9% | 9.8% | 8.9% | 8.9% | 8.7% | 8.9% | 8.9% | 19.5% | 8.9% |
| Ashleigh | 8.9% | 19.5% | 9.8% | 8.9% | 8.9% | 8.7% | 8.9% | 8.9% | 8.9% | 8.9% |
| Brittany | 19.5% | 8.9% | 9.8% | 8.9% | 8.9% | 8.7% | 8.9% | 8.9% | 8.9% | 8.9% |
| Coleysia | 8.9% | 8.9% | 9.8% | 19.5% | 8.9% | 8.7% | 8.9% | 8.9% | 8.9% | 8.9% |
| Jacy | 8.9% | 8.9% | 9.8% | 8.9% | 19.5% | 8.7% | 8.9% | 8.9% | 8.9% | 8.9% |
| Jessica | 8.7% | 8.7% | 22.0% | 8.7% | 8.7% | 8.5% | 8.7% | 8.7% | 8.7% | 8.7% |
| Kayla | 8.9% | 8.9% | 9.8% | 8.9% | 8.9% | 8.7% | 8.9% | 8.9% | 8.9% | 19.5% |
| Paige | 8.9% | 8.9% | 9.8% | 8.9% | 8.9% | 8.7% | 19.5% | 8.9% | 8.9% | 8.9% |
| Shanley | 9.8% | 9.8% | X | 9.8% | 9.8% | 22.0% | 9.8% | 9.8% | 9.8% | 9.8% |
| Simone | 8.9% | 8.9% | 9.8% | 8.9% | 8.9% | 8.7% | 8.9% | 19.5% | 8.9% | 8.9% |
Stay tuned for next week when the real strategy begins.
I'm interesting in learning more about how you go from a denominator of 3265920
ReplyDeleteto 608153 given that we know we have 2 out of 10 right in this scenario.
Alex, I really find this blog insightful and very helpful in understanding the probabilities of the show! But as with the previous commenter, I am also curious about how you got 608,153 possible arrangements. I've been trying to understand how you got to that point but I'm truly stuck and it's frustrating!
ReplyDeletethank you for this blog! growing up my dad encouraged me to be excited about math, but i lost my confidence in my abilities and decided i didn't enjoy it once middle school came around and he no longer had the time to help me with homework.
ReplyDeleteonly just recently as an adult i've been rebuilding that interest and finding fun contexts to learn about math topics has been super helpful for that!
speaking of, the line "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." made me laugh and reminded me of a time back in elementary school when i was learning probability for the first time and the textbook homework had a problem something like 'there's 5 lizards, 2 birds, 12 dogs, and 8 cats in the pet shop, what's the probability of a customer buying a bird?' which prompted a letter to my teacher from my dad complaining about how misleading that question was for kids just learning probability lolol
Why did the numbers for Jessica x Chris T and Shanley x Ethan increase?
ReplyDeleteHey! I would love to see your calculations! Im just really curious!
ReplyDeletebye
Alex how did you generate this table and values? extremely interested!
ReplyDelete