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Marvel's Birthday Research Paper

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Generalization: A generalization of this problem as well as its preceding problem (When is Cheryl’s Birthday) would be to connect these questions to the ideas of the birthday paradox, and an even bigger idea of the pigeonhole principle. In math, the pigeonhole principle states that if there are n items to be put into m containers, with the criteria that n > m, then at least one container will hold more than one item. (Herstein, 1964) Another way to explain this principle in a more quantitative way would be to say that for natural numbers k and m, if n = km + 1 objects are distributed among m sets, then the pigeonhole principle asserts that at least one of the sets will contain at least k + 1 objects. (Youtube) This pigeonhole principle can …show more content…
Using the pigeonhole principle if we do not count leap year then with 366 people in the room there is guaranteed to be a pair of people who share the same birthday as there are 365 possible days to choose from. (Linz, 2006) The paradox also goes on further to explain that even with a group of 70 people there are 99.9% chances of 2 or more people sharing the same birthday and 50% chances with only 23 people, with these conclusions being made with the above assumption that we are going by a 365 day calendar excluding the leap year has an equally probable chance for a birthday. This also intuitively makes sense when one realizes that a comparison would be made really between every possible pair of people rather than the fixation on one individual and then solely comparing them to the …show more content…
For n = 0 and for n = 1 (and m > 0), that probability is zero because if there is only one pigeon then there cannot be a conflict of multiple pigeons in one whole. (Herstein, 1964) For n > m (more pigeons than pigeonholes) the probability is indeed one which goes hand in hand with the ordinary pigeonhole principle. However, even if the number of pigeons do not exceed the number of pigeonholes n < m, then due to random chance of the assignment to pigeons to pigeonholes there can be a great chance that a clash will indeed occur. For example, if 2 pigeons are randomly assigned to 4 pigeonholes then there is a 25% chance that at least one pigeonhole will have more than one pigeon, for 5 pigeons and 10 holes, that probability is 69.76%; and for 10 pigeons and 20 holes it is about 93.45%. If the number of holes stays fixed, there is always a greater probability of a pair when you add more pigeons. (Wikipedia) Connecting this principle back to Cheryl's age and birthday we can say that not only with 144, but with all non prime numbers we can use this principle to help find the percentage of how many repeating factors we may get when determining how to multiply

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