Sunday, February 27, 2011

Faking It On Your Wedding Day

Earlier this month we wrote about our love of podcasts and just last week I was listening to Japan: A Friend In Need from the BBC Documentaries Archive. Here I was in the month of love, listening to a podcast on the subject and I found math in an unexpected place.

The documentary is about an agency in Japan that supplies fake people, or actors I suppose. In particular, this agency will supply people to fill out your side of a wedding. In the given example, we met a young man whose parents were deceased and his siblings were astranged, such that he only had two friends to attend his wedding. So as to keep up appearances, unbeknownest to the bride, he hired parents, friends and relatives. All told, 30 people at his wedding were fake, costing him something like £3,000, equal to his recent redundancy compensation.

The agency claims never to have been caught, and they say that they "research their assignments assiduously", but it got me wondering just how long you could operate such a service without getting caught. How many weddings could you do before a repeat guest noticed that they had seen one of your actors at a wedding before?

The first wedding is simple, and guaranteed to go off without a hitch, but what about the second? Suppose every wedding has on average 30 guests from each family. In the second wedding we need all 30 people to not be from the 30 in the previous wedding. Still pretty easy in a country of 127 million. But what about the 30th wedding when there are 900 previous guests out there in the population? Things are still looking pretty good, but the probabilities are starting to pile up in a similar way to the phenomenon that means that in a group of 23 people there's a 50% chance that two will have the same birthday.

So given a constant wedding size of 60, 30 real and 30 fake, what is the probability that this is the wedding that breaks us? This is the same as the probability that one or more of today's guests attended a previous wedding. This is the same as one minus the probability that none of today's guests attended a previous wedding. For wedding n and a population p:
Assuming 127 million people in Japan...
  • For wedding 1, it's a sure bet as nobody has attended a previous wedding.
  • For wedding 2, we face only a 0.0011% chance of getting caught.
  • Even for wedding 100 our risk is only a 0.11% chance. No problem!
But wait, the above probabilities are conditional probabilites. Our chance of getting caught at wedding 100 given that we got to wedding 99 is 0.11%. What is our chance of getting to wedding 99? This is the the probability that we didn't get caught in one or more of the previous weddings, the probability of a perfect record. Mathematically our chance of getting to and past wedding n is:
  • For wedding 1, it's a sure bet.
  • For wedding 2, it's 99.99%
  • For wedding 100, it's 94.58%.
  • For wedding 500, it's 24.57%.
Even though by the time we get to wedding 500, ony 15,000 people in Japan have been to weddings with our staff, we would be lucky to have made it that far.

If we started this agency today, on average how long can we expect to go before we get caught? Now I'm not going to bother expressing that mathematically, but hacking at it with Excel numerically, I can tell you that it comes to roughly 374. If we were to start such an agency today under such conditions and such assumptions, we would on average expect to do 374 weddings before getting caught.

So I think the moral of the story is, if you're looking to hire fake people for your wedding, you're doing alright, but if you're looking to run a business doing it, you might want to reconsider. Then again, if we're looking for morals in this story, honesty might come first.

1 comment:

Matt said...

I think you've made some assumptions here that are non-trivial. You assume that no honest attendees (i.e. not actors) can attend more than one wedding. You also assume that the 30 actors are always the same.

If you wanted to correct this, I suppose you would have to define some measure of turnover in the firm's staff of actors or a sub-population of actors that you, as management could draw from. Such an adjustment has to mean your odds of getting caught go down but I'm not sure how much... On the side of the honest attendees, I think it is reasonable to expect that any person who has attended a wedding is no less likely to attend again than any other person in the population. This would essentially change the model to one where you sample with replacement, instead of without, if I'm not mistaken. Maybe these are small details but it would be interesting to see the sensitivity of your calculations to these factors. Love your blog, keep it up.