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The Monte Carlo method
After looking at some of the ways that you can manipulate random numbers it is time to return to our main subject – the Monte Carlo method.
It isn’t difficult to think up serious uses of random numbers in a computer. For example, to find out how many pumps were needed at a petrol station you might simulate the situation. All you would need to do is write a program that allowed cars to join queues at the pumps with the correct statistical distribution and rate. You would also have to make sure that each fill of the tank took an appropriate amount of time and all the other random delays were included in the model. At the end of the simulation the program would provide you with a figure for the average queue length.
Notice that a simulation doesn’t give you the exact answer fixed and unchanging. It’s just an estimate of the answer and if you run the simulation again you will get a different answer. In practice of course you can put a probability estimate that places the answer in a given range with a high probability. For example, the queue might be estimated at between 8 and 11 with 90% certainty.
If you think too deeply about this you can start to get worried. After all if there is a 90% chance that it is in this range but there is a 10% chance that it is outside of this range! In fact there is a probability that it is any value you care to think of!
Of course in practice you don’t get worried because after throwing a coin 100 times and finding that the estimate of it coming down head is around .5 you are very happy to accept this as a reliable estimate of the value.
Simulations are somehow an “obvious” use of randomness to work something out, but it can get a little stranger than this because some problems that seem to have nothing to do with randomness can be solved by converting them into a form that can be simulated.
Mostly when you think about working out problems using a computer you think of writing a program to solve something exactly. For example, if you want to find the area of an irregular shape then the usual way of doing this is to divide the shape up into small rectangles, work out the area of each one and add the whole lot up. Of course the answer isn’t exact but if you make the rectangles smaller and smaller you get closer and closer to the correct answer.
The smaller the rectangles the more accurate the area
You may be wondering what this has to do with random numbers?
The answer is that this process of finding areas gets really tedious when you start to extend it to finding volumes and their higher dimensional counterparts.
Finding areas, or integrating functions, is a very important tool in physics and engineering and it was the area under a particular curve that was holding up the A bomb project. The computers of the day just couldn’t do the sums fast enough. Then John Von Neumann thought of a simpler way of doing the same job using random numbers.
To see how this works just think about firing a gun randomly at a target. The number of times you actually hit the target is going to be proportional to the target’s size or area. This is all there is to the Monte Carlo method.
For example, suppose we want to find the area of a circle. All we have to do is generate random (x,y) points and count the number that hit the circle and the number that don’t. If you do this enough times then the ratio slowly settles down to the ratio of the area of the circle to the background area.
You might think that converting this perfectly deterministic calculation into a game of chance is a crazy idea but you have to remember that generating random pairs is cheap and so is testing for a hit. Compared to the computational difficulty of dividing the shape up into little strips it is a very cheap way to work out an area and what is more the pay off gets better as you increase the number of dimensions you are working with. This is the reason that the area under an important function in the atomic bomb problem was worked out using random numbers.