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Finding the probability repeats in finite sequences

1. What is the probability that out of 3 people, 2 were born in the same month.

2. What is the probability that a seven digit phone number has 1 or more repeats.

3. What is the probability that given 5 letters selected randomly from the alphabet, none is repeated?

4. What is the probability 2 of more randomly selected senators have the same birthday (100 senators).

Solution Summary

The solution comprises 1 1/2 pages written in Word with equations in Mathtype. The first 3 questions are answered using basic probability and logic. The final question, known as the Birthday Problem, is more subtle. Whilst logic and basic probability do give an exact expression for the answer, calculators are not able to calculate this value (the numbers get too big) and more complicated techniques have to be employed (approximating first as a Binomial then as a Poisson distribution, or writing an iterative computer program) to obtain a numerical solution.

Full explanation is given for each part of the solution.

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