Chevalier de Mere's puzzle and other questions of probability
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1. Chevalier de Mere's puzzle (Scandal of Arithmetic)
Consider two experiments:
a. Roll a fair die 4 times. Record the number on top.
b. Roll a pair of fair dice 24 times, record the pair on top.
For experiment a, find the probability of event A: at least one 6. For the second experiment, find the probability of event B: at least one pair of 6s.
2. Consider the communication network below. Suppose Ai denotes that link i is operational, P(Ai) = pi and that the operation of different links is independent.
a. Find the probability that there is communication between points X and Y.
b. Find the conditional probability that link 5 is operational, given that there is communication between X and Y.
3. An engineering system consisting of n components is said to be a k-out-of-n system (k <= n) if the system functions if and only if at least k of the n components function. Suppose all components function independently of each other.
a. If the ith component functions with probability pi, compute the probability that a 2-out-of-4 system is functioning.
b. Repeat (a) for a 4-out-of-5 system.
c. Repeat (a) for a k-out-of-n system, when all pi's are equal to p.
4. Suppose n letters and n envelopes are addressed to n distinct individuals. A lazy individual randomly stuffs letters in envelopes, one per envelope. Let An be the event that at least one letter is in correct envelope.
a. Find P(An) (Hint: use inclusion/exclusion principle)
b. What happens to the above probability as n goes to infinity?
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