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Tirimwb 2022-07-14

Probability of 1 billion monkeys typing a sentence if they type for 10 billion years
Suppose a billion monkeys type on word processors at a rate of 10 symbols per second. Assume that the word processors produce 27 symbols, namely, 26 letters of the English alphabet and a space. These monkeys type for 10 billion years. What is the probability that they can type the first sentence of Lincoln’s “Gettysburg Address”?
Four score and seven years ago our fathers brought forth on this continent a new nation conceived in liberty and dedicated to the proposition that all men are created equal.
Hint: Look up Boole’s inequality to provide an upper bound for the probability!
This is a homework question. I just want some pointers how to move forward from what I have done so far. Below I will explain my research so far.
First I calculated the probability of the monkey 1 typing the sentence (this question helped me do that); let's say that probability is p:
P ( Monkey 1 types our sentence ) = P ( M 1 ) = p
Now let's say that the monkeys are labeled M 1 to M 10 9 , so given the hint in the question I calculated the upper bound for the probabilities of union of all P ( M i ) (the probability that i-th monkey types the sentence) using Boole's inequality.
Since P ( M i ) = P ( M 1 ) = p,
P ( i M i ) i = 1 10 9 P ( M i ) = 10 9 p = 10 9 p
Am I correct till this point? If yes, what can I do more in this question? I tried to study Bonferroni inequality for lower bounds but was unsuccessful to obtain a logical step. If not, how to approach the problem?