The Martian year is equal to 669 Martian days. Find the probability that there a

Question

The Martian year is equal to 669 Martian days. Find the probability that there are (at least) two Martians with the same birthday in a group of (a) 2 Martians (b) 3 Martians f (c) 15 Martians Find the minimal size of a group such that the probability p that there are (at least) two Martians with the same birthday in such a group is greater than 0.5: ____ You come to a cafe to grab a coffee at a random time between 5 pm and 6 pm and spend 20 minutes in there. Independently, your friend comes to the same cafe at a random time between 5 pm and 6 pm and spends 20 minutes in there. Find the probability that you meet each other _____

Explanation / Answer

P(two people share birthday) + P(no two people share birthday) = 1
P(two people share birthday) = 1 - P(no two people share birthday).

So, THe probability that two person share the same birthday = ?

Let say one martian has a date out of 669 days so the other one will have the same date have the probability = (669/669) * (1/669)

(b)Now we have to calculate that out of 3 martians two have same birthdate.

so first we will calculate that all 3 have different birthdays which is

P(all 3 have different birthdays) = (669/669) * (668/669) * (667/669)

P(one pair will share birthday = 1 - (669/669) * (668/669) * (667/669) =0.0055

(c) Now out of 15 martians

P (all n have different birthdays) = 669Pn / 669 n

P (2 of them have same birthdays) =1 - 669Pn / 669 n = 1 -  669P15 / 669 15   

NUmber of groups such that probability p is more than 0.5 for that number let say it is n

1 - 669Pn / 669 n > 0.5

669Pn / 669 n < 0.5

2 * 669Pn <  669 n

we can calculate it by exponential approximation

1 - e-n(n-1)/(2* 669) >0.5

e-n(n-1)/(2* 669) < 0.5

n(n-1)/ 1338 < 0.69314

n (n-1) > 927

n > 30

so for more than 30 martians probability will be more than 50%.

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