I don't follow you math... it looks like to me that the odds that a 100 monkeys can produce this 215 page document over 1 billion years is 1 in 2.12 x 10^292.. Your original probability was 1 in 5.36*10^307 for a single monkey producing a single document. That is a lot of zeros...
5,360,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
The odds of a random series of spaces and letters 215 words/spaces long forming a specific combination of letters and words is 27^215. Its actually a bit less because some are repetitions. But for simplicity's sake, we will ignore them.
The number 27 is used because there are 26 letters in the alphabet and 1 space. I am ignoring grammatical punctuation because it makes the numbers far too big for normal programs to manage. It would limit my calculations to less than 90 words, making it useless. the number 215 was chosen because that is the limit for my calculator. Anything more comes out to infinity.
27^215=5.36*10^307.
There are 5.36*10^307 possible outcomes with 27 characters (26 letters+spaces)
The odds of an event occurring is (desired outcome)/(possible outcomes)
There are 2 desired outcomes. One forward, and one backwards
So, the probability of a random paper produced matching any document of your choice is
2/5.36*10^307.
So the odds of a single monkey banging away on a type writer is so low, it is effectively zero. But that monkey is not stopping. It will hit that keyboard 215 times, then start all over again. I assume it hits the keyboard at a rate of 90 letters per minute. That monkey will produce a document every 2.36 minutes. I rounded that to 2.5 for convenience. (That decreases the odds of success, so it does not twist things in my favor). We can't be cruel, so that monkey bangs away on that keyboard for 8 hours a day, 7 days a week. No weekends for them. Every day, that monkey will produce 192 documents. now 192/5.36*10^307 is still effectively zero. But we have 100 monkeys banging away on a keyboard. So we actually have 19,200 documents a day.
This is a binomial probability. Either the document is a success, or it is a failure. They are disjoint (you can't succeed and fail at the same time) and independent of one another (assuming the monkeys do not copy each other). Disjoint means the two outcomes are mutually exclusive. Independent means the outcome of one does not effect the outcome of another.
So we can calculate the probability of this event not happening.
There are 2 outcomes which result in success. And (5.36*10^307)-2 outcomes which result in failure
So the odds of 1 paper failing are
((5.36*10^307)-2)/5.36*10^307.
We subtract 2 because that is the number of outcomes which gives us success. The result is pretty much zero.
But the odds of 19,200 randomly produced papers all failing is
(((5.36*10^307)-2)/5.36*10^307)^19200
Why did we raise the probability of 1 failing to the power of 19,200?
If events A and B are independent of one another, the odds of both occuring are P(A)*P(B), probability of A multiplied by probability of B. There are 19,200 papers, all independent. Each paper has the same chance of failing so we take the probability of 1 paper failing, and raise it to the power of 19,200.
The probability of all papers produced by monkeys in one day failing is
0.9999999999978684
Because the two outcomes of a paper are disjoint, you can subtract this probability from 1 to get the odds of success.
Those odds are extremely low, effectively zero. A snowball's chance in hell. But that is only for one day. Since we know how many papers the monkeys produce in one day, we know how many papers the monkeys will produce in a year. These guys will churn out 700,800 papers in a year. Now we have a general formula we can use.
(((5.36*10^307)-2)/5.36*10^307)^Papers.
You said 1 billion years correct?
Since we know that the monkeys will produce 700,800 papers a year, we know how many papers they will produce in 1 billion years. The monkeys will produce 7,008,000,000,000,000 papers in 1 billion years
We plug that into our nifty little formula
(((5.36*10^307)-2)/5.36*10^307)^70,080,000,000,000,000
The odds of all these papers failing to produce a success is .459.
1-.459=.541.
The odds of those monkeys making a paper of any type is 54.1% Shakespeare, whatever.
This is based off of
letters and spaces. I bolded that for a reason. 215 letters and spaces. No commas, hyphens, quotes, etc. It will be a short paragraph. You need a beefy calculator or specific programs to go beyond 310ish decimal places.
edit: D'oh!
I dropped a zero on the number of papers produced. There should be another on the end. Unfortunately, that number screws up my old ti-89 and I am getting a probability of 1, which is impossible. If anybody has a powerful enough calculator, could you please enter this in?
(((5.36*10^307)-2)/5.36*10^307)^70,080,000,000,000,000
In scientific notation that is 7.008*10^16