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Leap Year Probability

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by shankar.ashwin » Fri Sep 23, 2011 11:03 pm
How many randomly assembled people are needed to have a better than 50% probability that at least 1 of them was born in a leap year?

A. 1
B. 2
C. 3
D. 4
E. 5
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by knight247 » Sat Sep 24, 2011 12:00 am
Lets assume there are N people

Out of these N ppl, the probability that any one of them was born in a leap year is 1/4
Probability that the same person is not born in a leap year is 1-1/4=3/4
So the probability that all N ppl are not born in a leap year is (3/4)^N

Now,
1-(3/4)^N is the probability that atleast one of them is born in a leap year.
It needs to be greater than 1/2 so we use the inequality

1-(3/4)^N>1/2
1/2>(3/4)^N

(3/4)^N<1/2

Now try plugging in different values of N from the answer options
(A)if N=1 then
3/4>1/2 so NO
(B)N=2
9/16>1/2 so NO
(C)N=3
27/64<1/2 so YES
(D)N=4
81/256<1/2 so YES
(E)N=5
243/1024<1/2 so YES

I guess the question should have asked what is the minimum number of ppl. The question doesn't seem to be properly worded. Anyway, if it is the minimum number of ppl then the answer is C
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by shankar.ashwin » Sat Sep 24, 2011 12:22 am
Agree the question could be worded better. But what you did is perfect, that what the question asks.

The OA is C
knight247 wrote:Lets assume there are N people

Out of these N ppl, the probability that any one of them was born in a leap year is 1/4
Probability that the same person is not born in a leap year is 1-1/4=3/4
So the probability that all N ppl are not born in a leap year is (3/4)^N

Now,
1-(3/4)^N is the probability that atleast one of them is born in a leap year.
It needs to be greater than 1/2 so we use the inequality

1-(3/4)^N>1/2
1/2>(3/4)^N

(3/4)^N<1/2

Now try plugging in different values of N from the answer options
(A)if N=1 then
3/4>1/2 so NO
(B)N=2
9/16>1/2 so NO
(C)N=3
27/64<1/2 so YES
(D)N=4
81/256<1/2 so YES
(E)N=5
243/1024<1/2 so YES

I guess the question should have asked what is the minimum number of ppl. The question doesn't seem to be properly worded. Anyway, if it is the minimum number of ppl then the answer is C
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by Brent@GMATPrepNow » Sat Sep 24, 2011 6:23 am
Aside: I think it's unlikely that you'd ever see a GMAT question involving leap years.
The main reason is that P(born in leap year) doesn't equal 1/4 (it's close to 1/4, but not equal)

From wikipedia (https://en.wikipedia.org/wiki/Leap_year):
The Revised Julian calendar adds an extra day to February in years divisible by four, except for years divisible by 100 that do not leave a remainder of 200 or 600 when divided by 900.

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by shekhar.kataria » Sat Sep 24, 2011 11:11 am
shankar.ashwin wrote:How many randomly assembled people are needed to have a better than 50% probability that at least 1 of them was born in a leap year?

A. 1
B. 2
C. 3
D. 4
E. 5
Hi

Can anyone please confirm the source. Is this a real GMAT type question?? I see many questions which i feel are beyond the scope of GMAT.
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by shankar.ashwin » Sat Sep 24, 2011 11:23 am
I got it in a practice test, but as Brent says maybe Leap Year questions may not appear in the GMAT, but that said the probability part of this question is pretty straightforward and IMO can be expected in the GMAT especially on high difficulty quants questions. I think experts would agree too.
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by Brent@GMATPrepNow » Sat Sep 24, 2011 11:39 am
shekhar.kataria wrote: I see many questions which i feel are beyond the scope of GMAT.
This is a common problem.

Sometimes people/companies like to save time by re-purposing questions they created for a different test, but the questions don't fit the GMAT curriculum.

Other times, we get students using the GMAT forums to get help with their homework in Calculus, Stats, Number Theory etc.

Takeaway: you have to be careful when looking for legitimate practice questions. Otherwise you may waste time needlessly worrying about concepts that the GMAT does not test.

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by Brent@GMATPrepNow » Sat Sep 24, 2011 11:40 am
shankar.ashwin wrote:I got it in a practice test, but as Brent says maybe Leap Year questions may not appear in the GMAT, but that said the probability part of this question is pretty straightforward and IMO can be expected in the GMAT especially on high difficulty quants questions. I think experts would agree too.
Agreed - the probability component of this question is within the scope of the GMAT.

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