Study the special case of the small world problem called the birthday problem (h
ID: 3204319 • Letter: S
Question
Study the special case of the small world problem called the birthday problem (http://en.wikipedia.org/wiki/Birthday_problem). Disregarding the possibility of a February 29 birthday, suppose a randomly selected individual is equally likely to have been born on any one of 365 days. a. If eight people are randomly selected, what is the probability that all have different birthdays? b. If eight people are randomly selected, what is the probability that at least two have the same birthday? c. Create a table for k people being randomly selected, from k = 2 to k = the smallest value for which there is at least a 50-50 chance that at least two people will have the same birthday. (you should use Excel)
Explanation / Answer
probability that all have different birthdays =P(first one has 365 choices ,second 364, third 363,....)
=(365*364*363*.362*361*360*359*358)/(365)8 =0.00093
hence probability that at least two have the same birthday =1-P(none of have same birthday)
=1-0.00093 =0.99907
from above probabilty for at least 2 people share same bth day =1-(365!/(365-k)!)/(365)k
therefore below is the table
from above at k=3 there is 50-50 chance/.
K P(k) 2 0.003 3 0.008 4 0.016 5 0.027 6 0.040 7 0.056 8 0.074 9 0.095 10 0.117 11 0.141 12 0.167 13 0.194 14 0.223 15 0.253 16 0.284 17 0.315 18 0.347 19 0.379 20 0.411 21 0.444 22 0.476 23 0.507