Yes this is about the movie “Identity” but it is not a Cafe Society question. Also, I don’t think I’m giving away any plot secrets so I don’t think spoiler boxes are necessary.
Due to a storm, 10 people have been randomly brought together in an out of the way motel in Nevada. Eventually, it is discovered they all have the same birthday - May 10th (not the same year though). Someone guesses the probability of that happenng are 10 trillion to one. At first, I thought 10 trillion to 1 was a rather high estmate - wow was I wrong.
If you start with 1 person, they will have a birthday and the probability of that happenng is certainty or 1. (It was not specified that they all had to have a May 10th birthday).
So, the second person would have to have the same birthday as the first person. Now the odds become 1 in 365. (Yes I am ommiting leap year birthdays out of the calculations and this calculation and I’m assuming that birthdays are scattered evenly throughout a calendar year). So when you reach the tenth person the probability of all 10 having the same birthday are 1 in 365[sup]9[/sup] or 1.15 x 10[sup]23[/sup] or about 1 in 115 sextiliion !!!
So yes there is a question in here. Are my calculations right? I’m still surprised the probability is that unbelievably high.
I have not seen the movie. I agree with your calculations.
For 10 people to be born in the same day of the month (but allowing different months) would be roughly 1 in 30^9, which interestingly enough is about 19 trillion to 1. Perhaps they did the math wrong, or more likely felt that 10 trillion was the highest number that viewers would come close to comprehending.
Given that birth date distribution is a bit lumpy, I’d guess that the “real-world” probability could be up to an order of magnitude higher than 10[sup]-23[/sup], but still far lower than suggested in the movie. Nothing wrong with your math, besides the fact that it yields a non-alliterative answer.
This is almost surely the reason. The script-writer probably didn’t bother doing the math at all, and in production they got someone to check that it’s really unlikely. Since most people still don’t hear about trillions that often – “Billions and Billions” is even pushing it – unless the character was a mathematician or scientist it would sound very forced to use the term “sextillion”.
A related case is the old ad copy for the Rubik’s cube advertizing “more than three billion” possible states. The real number is 43,252,003,274,489,856,000 – more than fourteen billion times as many. It’s like saying “McDonalds: more than 2 served.” Still, people at the time were (and many still are) boggled by the idea of a billion, and “43 quintillion” would just go straight over their heads.
Thanks for verifying my calculations. Granted, it is a relatively easy probability problem, but that number is just so huge I had a feeling I was doing something wrong. Guess not.
The ratio of 1 part in 1.15x10[sup]23[/sup] is equivalent to 1 foot in 3.7 million light years!!! (More than 1.5 times the distance to the Andromeda Galaxy.)
Just as with AWB’s example, a good rule to follow when dealing with probability problems is never trust your intuition, feelings, etc. You will usually end up wrong.
Here’s a good example of common sense, gut feelings, intuitions, etc sending you down the wrong mathematical path. There are 36 possible outcomes when 2 dice are thrown and only 1 of those outcomes is a 12. (That part is true).
A gambler named “Fat the Butch” was offered a bet that if he rolled a pair of dice 21 times and got a 12, he would win $1,000 but if he didn’t get a 12 in 21 rolls, he had to pay $1,000.
Well the gambler figured there’s 1 way in 36 to get a 12. So after 18 rolls, the odds are 50-50 and so with 21 rolls, the odds are well in your favor. After several hours of this, Fat the Butch realized there was something wrong with his probability calcualtions because he lost $49,000 and decided to call it quits for this little game. (True story - circa 1950)
Look at it this way: The probability of NOT getting a 12 (boxcars) on any given roll are 35/36. The probability of getting zero boxcars in 21 rolls is thus (35/36)^21 which works out to be a hair over 55% (55.34%).
The odds are certainly against Fat The Butch but not by a huge amount - expected loss is 10% of the wager each time. To be down $49k it’d take (on average) about 490 wagers. Ole’ Fat musta had a lot of time on his hands!