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.

...Find the expected value at each age ... Thus Binomial distribution is the discrete probability distribution of the number of successes ... If we repeat this experiment. ...

... the worst possible outcomes when the probability is favorable ... c. the joint probabilities must total less than ... penalties are determined by a. finding the largest ...

... approach in the third section to ﬁnd probabilities. ... The letter P here represents permutations, not probability. ... Sequencing Eleven drugs have been found to be ...

... c. the joint probabilities must total less than 1. d ... deviation) = 5. Z = 2, for this we find from the table of normal distribution the probability for the ...

... b. Find the expected value of the number of tests taken. ... the values of a random variable with the probabilities of those ... Answer: da probability distribution. ...

... Do not report statistics found on a ... plots, spread/level plots, probability plots, spectral ... permutation) distributions, Neymanian repeated-sampling randomization ...

... right with the project that should be repeated with the ... team only which can decide the probability of its ... assist each other if any individual finds problem in ...

...Probability of remembering something related to number of ... Sperling found that subjects could accurately recall ... recallthe ability to repeat short series of ...