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all. I'm working on the following probability problem.

Birthday problems: 50 randomly selected students end up in a class, 365 days in a year. Find P for the following events.

-i- A is the event that 3 randomly chosen students have the same birthday.

-ii- B is the event that the tallest, second and third tallest student in the class each has the same birthday (assume no two people in the class are of exactly the same height).

-iii- C is the event that the tallest, second and third tallest student in the class each has a different birthday (assume no two people in the class are of exactly the same height).

-iv- No one in the class was born in the month of August.

-v- Every person in the class was born in one of the four Fall months (September through December).

I'm a little confused about parts (ii) and (iii). For problem (i), I understand that $$ |\Omega|= 365^3 \, \Rightarrow P(A) = 1-\frac{365*364*363}{365^3} =0.0084$$ But for parts (ii) and (iii), I don't quite understand how to incorporate the sizes of each individual. Do their sizes matter, or can I just treat this problem the same I would for part (i)?

Thanks!

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  • $\begingroup$ The answer to the first part should be $365\times\frac{1}{365^3}$ $\endgroup$
    – David Quinn
    Commented Jan 28, 2018 at 22:56
  • $\begingroup$ what you have for (i) is the probability that the birthdays of three random students are not all different $\endgroup$
    – WW1
    Commented Jan 28, 2018 at 23:16

2 Answers 2

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Hint...the result you get for part i (which you need to reconsider, by the way) is the same for any three students. Their respective heights are irrelevant.

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  • $\begingroup$ I initially got David Quinn's answer (from the comments), but a quick Google search suggested I do it the way I did it above. So, then, part (iii) is simply 1-P(A)? $\endgroup$
    – NoVa
    Commented Jan 28, 2018 at 23:15
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It is not specified in the problem statement, but from everyday experience it seems not too far-fetched to assume that birthday (without year) and height are independent. (Though once you start thinking about it for a longer while, you may arrive at the assumption that they are slightly dependent; e.g., if all students are born in the same year, those with January bithday are older, hence expected taller than December borns). Throwing these slight doubts overboard, it is therefore that picking specific people by their height is the same as picking random people.

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