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Birthdays---Challenging

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by knight247 » Mon Feb 20, 2012 10:18 am
Two people have birthdays on different dates of the same 30-day month. Assuming the dates of the birthdays are independent of one another, what is the probability the birthdays fall on the same day of the week each year?
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by Mike@Magoosh » Mon Feb 20, 2012 11:58 am
Hi, there. I'm happy to provide a solution to this. :)

In a 30 day month, the following dates will always land on the same weekday:

Group A: 1-8-15-22-29
Group B: 2-9-16-23-30
Group C: 3-10-17-24
Group D: 4-11-18-25
Group E: 5-12-19-26
Group F: 6-13-20-27
Group G: 7-14-21-28

The thirty days break down into two groups of 5 days, and five groups of four days.

P(birthday on same day)
= P((A and A) or (B and B) or (C and C) or (D and D) or (E and E) or (F and F) or (G and G))
= (5/30)*(5/30) + (5/30)*(5/30) + (4/30)*(4/30) + (4/30)*(4/30) + (4/30)*(4/30) + (4/30)*(4/30) + (4/30)*(4/30)
= (25 + 25 + 16 + 16 + 16 + 16 + 16)/900
= 130/900 = 13/90

That's the probability of the two people having their birthdays on the same weekday in a 30-day month.

Please let me know if anyone reading this has any questions.

Mike :)
Magoosh GMAT Instructor
https://gmat.magoosh.com/
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