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- The movie 21 is a black jack oriented drama based on the non-fiction book 'Bringing Down the House', a true story about how a group of MIT students won millions of dollars betting on black jack in Las Vegas by playing a system and counting cards. The movie stars Jim Sturgess, Kate Bosworth,.
- An impressive scene from 21 - The Movie.http://www.imdb.com/title/tt0478087/.
by Jeff Moehlis
The movie '21' isthe story of MIT students who 'count cards' to improve their probabilityof winning the card game Blackjack at casinos. Not surprisingly, this movie has a lot of mathematics in it. Most obvious is the 'counting of thecards', which is based on the techniques published in Edward O. Thorpe's1962 book 'Beat the Dealer'. Discussions of the method and mathematicsof 'card counting' are described on various other websites. On this website, you can learn about other mathematical ideas which appear in the movie. I hope that this increases your enjoyment of the movie andperhaps teaches you some mathematics!
The Fibonacci SeriesIn '21', when Ben Campbell (played by Jim Sturgess) is celebrating hisbirthday, the cake says
0, 1, 1, 2, 3, 5, 8, 13, ...
These are the first terms in the Fibonacci Series, which was used as an example in the book Liber Abaci published in 1202 by Leonardo Fibonacci.This is obtained by first writing the numbers '0, 1', then defining eachsubsequent number as the sum of the previous two numbers in the series. Thus, thethird number in the series is 1 = 1+0, the fourth number is 2 = 1+1,the fifth number is 3 = 2 + 1, etc. The next number on the cake wouldbe 21=13+8, for Ben's 21st birthday. Clever, huh? (Hmmm, does '21'refer to Blackjack or Ben's age?) Ben will have towait until he is 34 = 21+13 for his next 'Fibonacci birthday'.
One can define other Fibonacci Series by specifying different numbers inthe first two slots. For example, the Fibonacci Series starting with'2, 5' is
2, 5, 7, 12, 19, 31, 50, ...
The Monty Hall ProblemConsider the following variation of the final round of the classic TVgame show Let's Make A Deal:
There are three doors, and behindone of them is a car, while behind the other two are goats. If you choosethe door with the car behind it, you win the car. Now, say you choose Door 1. The host MontyHall then opens either Door 2 or Door 3, behind which is a goat. (He knowswhat is behind each door, and neveropens the door with the car behind it.) Monty now gives you the choice:do you want to stick with Door 1, or switch to the other door. Whatshould you do? Or does it matter?
A similar question is posed to Ben Campbell (played by Jim Sturgess) by Professor Micky Rosa (played by Kevin Spacey) in the movie '21'. Withouthesitation Ben answers this correctly, which convinces Professor Rosa that Ben would be a good addition to their 'card countingteam'. Before reading on, try to answer this yourself.
One solves this problem by comparing the probability of choosing the car if youstick with your original choice to the probability of choosing the carif you switch after Monty opens the one door. Note that the car has an equal probability of 1/3 of being behind Door 1, Door 2, or Door 3.
First, suppose that your strategy is to stick with your original choice of Door 1. Then you only win if the car is behind Door 1, so thatyour probability of winning is 1/3.
Next, suppose that your strategy is to switch doors. We break this intothree cases:
So if your strategy is to switch doors, you win 2/3 = 1/3 + 1/3 of the time. (Remember,the probability is 1/3 that the car is behind any particular door.)Therefore, a better strategy is to switch doors - the calculated probabilities indicate that you are twice as likely to win if you do this!Ben's correct answer in the movie '21' indicates that he is a good personfor 'counting cards'. Not only does it show that he is clever,but it also demonstrates that he realizes that it is best to go with the choice whichmaximizes your probability of winning. This realization is essential tothe success of 'counting cards' for Blackjack.
In 1990, a similar question appeared in a letter to Marilyn vos Savant's Ask Marilyn column in Parade (which comes in some Sundaynewspapers). Marilyn gave the correct answer, but many readers(including mathematics professors) believed that this was incorrect.So don't feel too bad if you got it wrong when you answered it for yourself.But now you know!
You might recall from an algebra class that the solutions to theequation
are given by the quadratic formula
Suppose instead that you want to find a value for x which solvesthe general algebraic equation
f(x) = 0.
Such a value for x is called a root of f(x). Except for special choices of f(x), such as f(x) = a x2 + b x + cas above, one cannot find the roots using algebraic operations.
In the movie '21', Professor Micky Rosa (played by Kevin Spacey)is lecturing on the Newton-Raphson method for finding the rootsof f(x). This was developed independently by Isaac Newton and Joseph Raphson in the 1600's. The idea is to make a guess for a root of the equation (let's call it x0), then to use this guess to generate a value for x (let's call it x1) which is (hopefully) even closer to the root than the original guess. This isdone by drawing the tangent line to the function f(x) at x=x0, and takingx1 as the value for x at which this straight line goes through zero.(For those of you who know calculus, you will recognize that this tangentline is determined by the derivative of f(x).)By iterating this procedure over and over to generate x2, x3, etc, one (hopefully) obtains values which are better and better approximations to the root. I keep saying 'hopefully' because the Newton-Raphson Method isn't always successful, although it is more likely to be if you make a good initialguess. This figure illustrates the method:
This method was developed well before computers existed, but turns outto be ideal for implementation on a computer: one uses a loop to generate successive values of xn.
Counting Cards
A nice discussion of 'counting cards' for Blackjack is givenin this Wikipediaarticle .
Now let's count the number of times this page has been visited:
Theatre 21 Watch Online Movies
Visit Author Jeff Moehlis'website. Also, check out his music site music-illuminati.com.
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