Disregarding the possibility of a February 29 birthday, suppose a randomly selected individual is equally likely to have been born on any one of the other 365 days. a. If ten people are randomly selected, what is the probability that all have different birthdays? That at least two have the same birthday? b. With ?k ?replacing ten in part (a), what is the smallest ?k ?for which there is at least a 50-50 chance that two or more people will have the same birthday? c. If ten people are randomly selected, what is the probability that either at least two have the same birthday or at least two have the same last three digits of their Social Security numbers? [? ote: T he article “Methods for Studying Coincidences” (F. Mosteller and P. Diaconis, ?J. Amer. Stat. Assoc., ?1989: 853–861) discusses problems of this type.]

Answer Step 1 of 3 10 a) There are 365 possible list of birthdays P(all different)=P 10, 36536510 =0.883 P(at least two of them same)=1-0.883=0.117 Step 2 of 3 k b) the formula for P(at least two of them same)=1-(P k, 365365 ) By trial and error the probability equals 0.476 when k=22 the probability equals 0.507 when k=23 23 people have 50% chance will have the same birthday There 1000 possible 3- digit sequences to end SS Number P(at least two have the same SS number)=1-(P 10, 10001000 ) =1-0.956 =0.044