Math Discrete...

Birthdays

			Enter as appropriate using the values along Column B.
	n	n - 1	P(~same)		P(same)		¬ P(Same) is the complement of P(~Same)
¬ continue the values of n and n-1	2						¬ Note that we have two sets of merged cells... DO NOT UNMERGE EITHER SET
	3
	4

(c)2017 Second Wind Productions, LLC

(c)2017 Second Wind Productions, LLC

Here's the question: What is the minimum number of people that need to be in a room such that there is better than a 50% chance that at least two of them share the same birthday of the year? To answer this question, we have to figure out the probability that no two people in n (n > 1) have the same birthday. Subtracting that from one yields at least two people share the same birthday. Fill in the values down below the next text box, as instructed, and then answer the two below questions. This is worth 3.5 CPs (very generous, Mr. Ulrich!): At most 2 CPs for the first question, 1.5 CPs for the second question. 1) Assuming a standard 365-day year, what is the minimum number of people needed for there to be at least a 50% probability that at least two share the same birthday? You MUST perform your work below the second text box to get any of this first question's credit! Show no work in this text box! answer here  2) The Martian calendar is 687 Earth days long. Let's say that we have a colony on Mars, and that we have a unique date label for each of the 687 days. If possible, what is the minimum number of people we would need to have in that colony to guarantee 100% that at least two people share the same birthday in a Martian year? This is a thought experiment! Do not try to work it out, nor try to explain it! answer here 

Here is how we have to look at this problem...There is a 100% probability that you have your birthday, hence a 0% probability that you don't have your own birthday. But what is the probability that the guy next to you DOES NOT have your birthday? Of the 365 days available is a standard year (leap years complicate this too much), there are 364 opportunities for that guy to NOT have YOUR birthday. So, for n = 2, the probability that two people do not share the same birthday is: P(~same) = 364/365  0.9973 Pretty much assured that you two do not have the same birthday. What, then, is the probability that you two do share the same birthday? Since either you do or don't share the same birthday, then: P(same) = 1 - P(~same) = 1 - 364/365 = 1/365  0.0027 Let's go to n = 3 to find the pattern. Now not only do neither of the two guys next to you have your birthday, but THEY ALSO do NOT share the same birthday with each other. So, the first guy has 364 opportunities to not have your birthday, while the other guy has only 363 opportunities to have neither of your birthdays. So: P(~same) = (364/365)(363/365) = (364*363)/(365)2  0.9918 And P(same) = 1 - 0.9918 = 0.0082 a smidge better One more time for n = 4: P(~same) = (364/365)(363/365)(362/365) = (364*363*362)/(365)3  0.9836 So P(same) = 1 - 0.9836 = 0.0164  1.6% now we are starting to get someplace noticable . . . The numerator is determined by a Permutation; information on this is found at the right  The denominator is just higher and higher powers of 365; specifically, it follows 365(n-1). Combining these, we get the probability that two people do not share the same birthday  n > 1. Subtracting that from 1, therefore, gives us the probability that at least two people share the same birthday. Caution, this is NOT the same thing as having exactly 2, or exactly 3, or exactly 4, . . . people having the same birthday. As n increases in value, the probability of at least 2 carries all the lower values of n as well. Select Cells D66-G66 >> right click >> format cells >> number >> number >> set decimals to 4 places >> enter.

Select the cell directly underneath the label P(same) and enter the following: =PERMUT(364,C66)/365^C66. Reselect and adjust decimals as instructed above right. Select the cell directly underneath the label P(same) and enter the complement of the cell to its left. Then select Cells B66-G66 and drag down until P(same) > 0.5000. Finish by answering the questions above .