(Solution) - There are 10 000 prison inmates in a certain state Independently -(2025 Original AI-Free Solution)

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There are 10,000 prison inmates in a certain state. Independently of each other, and independent of their behavior on previous days, assume that, on a given day, a prisoner has probability p = 0.000001 of escaping.
a. What is the probability, on a given day, that none of the prisoners escapes?
b. What is the probability that the state has n consecutive days without any escapes?
c. Let X be the number of days until the next escape occurs. What distribution does X have?
d. How many days does it take until we are at least 99% sure that at least one prisoner has escaped in the state?
e. Focus attention on a specific criminal. What is the probability that he escapes during a 50-year period?
f. If 10,000 prisoners are imprisoned, each for 50 years, how many prisoners do we expect to escape?
g. There is a very intelligent criminal named Scar-Foot. Instead of having probability of 0.000001 of escaping each day, he has probability 0.0001 of escaping each day. What is the probability that Scar-Foot is able to escape during a 50-year period?