Essay Example on How many people if any in a room shared a Birthday

Introduction Have you ever wondered how many people if any in a room shared a birthday Well how many people do you think it would take to find two people who share the same birthday In fact if you were to get the birthdays of 23 random people there would be a 50 percent chance that two of them would share the same birthday Now if we have a bigger group such as 75 people there would be a 99.9 chance that at least two of them would have matching birthdays This whole concept is actually the birthday paradox also known as the birthday problem I became interested with the birthday paradox one day when my seventh grade math teacher had briefly explained it to us and sounded very simple After the birthday paradox was explained to us my teacher wanted to test it out It worked out perfectly because we had exactly 23 people in the room minus the teacher Once we wrote all of our birthdays on a note card our teacher collected them and started to organize them by day and month Once they were all set out we quickly discovered that two students in the class had the same birthday It all just seemed to be a coincidence 

At the time I didn't really understand the math behind it although the whole idea of the birthday paradox really fascinated me Now that I am older I am able to take matter into my own hands and further my knowledge of the mathematical concept behind the birthday paradox I will do this by conducting my own experiment and using the birthdays of my peers II Birthday Paradox Background The birthday paradox tells us that with a group of 23 people there is a 50 chance that two people will share the same birthday But how could this possibly even be accurate There seems to be several explanations why many find this to be a paradox So let's say there s 23 people in one room If one person from that group of 23 tries to compare their birthday with someone else it would then make for 22 comparisons since that leaves only 22 chances for people to share the same birthday When the 23 birthdays of people in the group are compared against each other there's more than the 22 birthday comparisons meaning the 1st individual would have 22 comparisons to make and because the 2nd individual had already been compared to the 1st there would be 21 comparisons to make and so forth Afterwards if you add all possible comparisons which is 22 21 20 1 the sum you would end up getting would be 253 comparisons Therefore this shows that in a group of 23 people there will be 253 possibilities to having a shared birthday III Proving the Birthday Paradox Experiment Materials that I will need to conduct this experiment will be 6 groups of 23 people 6 groups of 75 people 588 people total Notecards Procedure I will prove the birthday paradox with 12 different groups Half of them will contain 23 people while the other half contain 75 people It is being split up this way so that we can see if the birthday paradox is true 

With the 6 groups of 23 people will half of the groups have at least 2 people who have the same birthday What about the other 6 groups that have 75 people Will they have a 99.9 chance that 2 people share the same birthday After I will write the out every person's birthday on a notecard Once completed I will shuffle all the cards and start creating the groups of 23 and 75 After all the cards are distributed in their groups I will collect the data of the matching pairs Proving birthday paradox mathematically In the groups of 23 people there are 253 possible pairs that exist and we know this because With the 253 possible pairs you can find the probability of two people that won t share the same birthday When comparing probabilities with birthdays it can be easier to look at the probability that people do not share a birthday A person's birthday is one out of 365 possibilities excluding February 29 birthdays The probability that a person does not have the same birthday as another person is 364 divided by 365 because there are 364 days that are not a person s birthday This means that any two people have a 364 365 or 99 726027 percent chance of not matching birthdays As mentioned before in a group of 23 people there are 253 comparisons that can be made So we re not looking at just one comparison but at 253 comparisons Every one of the 253 combinations have the same odds 99 726027 percent of not being a match If enter 364 365 253 into your calculator it's shown that there s a 49 95 chance that the 253 comparisons don t have matches Because of that the chances that there are similar birthdays with the 253 possible comparisons is 1 minus 49 952 which equals to 50 048 So the additional trials that there are the closer the actual probability comes to 50 percent