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There are 50 students in our class. (1) What is the probability that Jack and Mike have the same birthday? (2) What is the probability that no one has the same birthday as the professor? (3) What is the probability that there are two and only two having the same birthday?

 Feb 28, 2020
 #1
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Assuming only 365 days in a year --

 

1)  Jack's birthday can be any day:  365/365

      Mike's birthday must be the same day:  1/365

      Final probability:  (365/365) x (1/365)

 

2)  The professor's birthday can be any day.

      The first student can have a birthday any day except for the professor's:  364/365

      The next student cannot match either of the two previous persons:  363/365

       Etc.

       Final probability:  (364/365) x (363/365) x (362/365) x ... x (315/365)

 

3)  This one is a guess:

      The first person's birthday can be any day.

      To have only two persons with the same birthday, 48 persons cannot match:

            (364/365) x (363/365) x ... x (317/365)

       The one that matches:  (49/365)

       and this person can be any of the 49:

       Final probability:   (364/365) x (363/365) x ... x (317/365) x (49/365) x 49

       But this is just a guess.

 Feb 28, 2020
 #2
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(1)   n=50; p = 1 - ((365 nPr n) / (365^n))=97.03735796%

 

(2)  n=51; p =  ((365 nPr n) / (365^n))=2.55680067%

 

(3) [50C2 x 365!] / [365^50 x (365 - 48 +1)!=0.00011393 09727%

 Feb 28, 2020
 #3
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No. (1) answer is any pair of 50 students and NOT a specific pair of  Jack and Mike !!

Guest Feb 28, 2020

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