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# Twenty people at a meeting were born during the month of Sep

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Intern
Joined: 22 Jun 2014
Posts: 3
Twenty people at a meeting were born during the month of Sep  [#permalink]

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03 Aug 2014, 13:52
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Difficulty:

95% (hard)

Question Stats:

33% (02:14) correct 67% (01:54) wrong based on 132 sessions

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Twenty people at a meeting were born during the month of September, which has 30 days. The probability that at least two of the people in the room share the same birthday is closest to which of the following?

(A) 10%
(B) 33%
(C) 67%
(D) 90%
(E) 99%
Math Expert
Joined: 02 Sep 2009
Posts: 49948
Re: Twenty people at a meeting were born during the month of Sep  [#permalink]

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13 Aug 2014, 07:39
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kennaval wrote:
Twenty people at a meeting were born during the month of September, which has 30 days. The probability that at least two of the people in the room share the same birthday is closest to which of the following?

(A) 10%
(B) 33%
(C) 67%
(D) 90%
(E) 99%

PROBABILITY APPROACH:

P(at least two of the people share the same birthday) = 1 - P(none of the people share the same birthday) =
$$= 1 - \frac{30}{30}*\frac{29}{30}*\frac{28}{30}*\frac{27}{30}*\frac{26}{30}*...*\frac{11}{30} = 1 - \frac{30!}{(30^{20}*10!)}\approx{0.99}$$. First person can have birthday on any day (30/30), the second on any but that day (29/30), the thrid on any but those two days (28/30), ...

Notice that the number we are subtracting from 1 is very, very small, so the final result will be very close to 100%.

COMBINATIONS APPROACH:

P(at least two of the people share the same birthday) = 1 - P(none of the people share the same birthday) =
$$= 1- \frac{C^{20}_{30}*20!}{30^{20}}=\frac{30!}{(30^{20}*10!)}\approx{0.99}$$. $$C^{20}_{30}$$ here is choosing 20 different days out of 30, 20! is the number of ways we can assign 20 people to those 20 days (by the way, we could write there $$P^{20}_{30}$$ there instead of $$C^{20}_{30}*20!$$, which is basically the same: choosing 20 out of 30 when the order of the selection matters) and the denominator ($$30^{20}$$) is the total number of way 20 people can have birthdays in September (each of them has 30 options).

Hope it's clear.
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Joined: 22 Feb 2009
Posts: 175
Re: Twenty people at a meeting were born during the month of Sep  [#permalink]

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03 Aug 2014, 14:51
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kennaval wrote:
Twenty people at a meeting were born during the month of September, which has 30 days. The probability that at least two of the people in the room share the same birthday is closest to which of the following?
(A) 10%
(B) 33%
(C) 67%
(D) 90%
(E) 99%

The probability that at least two people sharing the same birthday = 1 - the probability that none of them sharing the same birthday
A = The number of ways of none of them sharing the same birthday = 30P20 = 30!/(30-20)! = 30!/10! = 11*12*...*29*30
B = The total number of possible ways of 20 people born in September = 20*20*....*20*20 = 20^30 ( each day has 20 options)
A/B = the probability that none of them sharing the same birthday
since B is much greater than A, A/B may equal 1%
--> The probability that at least two people sharing the same birthday = 1 - 1% = 99%

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Re: Twenty people at a meeting were born during the month of Sep  [#permalink]

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03 Aug 2014, 14:54
at least two of the people = 1- no two people share the same bday
no two people share the same bday = (1st pick a day in the 30 days) * (2rd pick another day in the left 29 days)
= (1/30) * (29/29)
so, at least two of the people share differ = 1-(1/30) * (29/29) = 29/30 = 99%
Intern
Joined: 22 Jun 2014
Posts: 3
Re: Twenty people at a meeting were born during the month of Sep  [#permalink]

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04 Aug 2014, 12:27
I still don't get it. I thought it would be 1-(29/30)*(28/30). Does anyone have another way of figuring this out?
Manager
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Re: Twenty people at a meeting were born during the month of Sep  [#permalink]

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04 Aug 2014, 14:08
kennaval wrote:
Twenty people at a meeting were born during the month of September, which has 30 days. The probability that at least two of the people in the room share the same birthday is closest to which of the following?
(A) 10%
(B) 33%
(C) 67%
(D) 90%
(E) 99%

The probability that at least two people sharing the same birthday = 1 - the probability that none of them sharing the same birthday
A = The number of ways of none of them sharing the same birthday = 30P20 = 30!/(30-20)! = 30!/10! = 11*12*...*29*30
B = The total number of possible ways of 20 people born in September = 20*20*....*20*20 = 20^30 ( each day has 20 options)
A/B = the probability that none of them sharing the same birthday
since B is much greater than A, A/B may equal 1%
--> The probability that at least two people sharing the same birthday = 1 - 1% = 99%

could you explain the red part please?
Math Expert
Joined: 02 Sep 2009
Posts: 49948
Re: Twenty people at a meeting were born during the month of Sep  [#permalink]

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13 Aug 2014, 07:41
kennaval wrote:
Twenty people at a meeting were born during the month of September, which has 30 days. The probability that at least two of the people in the room share the same birthday is closest to which of the following?
(A) 10%
(B) 33%
(C) 67%
(D) 90%
(E) 99%

The probability that at least two people sharing the same birthday = 1 - the probability that none of them sharing the same birthday
A = The number of ways of none of them sharing the same birthday = 30P20 = 30!/(30-20)! = 30!/10! = 11*12*...*29*30
B = The total number of possible ways of 20 people born in September = 20*20*....*20*20 = 20^30 ( each day has 20 options)
A/B = the probability that none of them sharing the same birthday
since B is much greater than A, A/B may equal 1%
--> The probability that at least two people sharing the same birthday = 1 - 1% = 99%

It should be 30^20, instead of 20^30: each out of 20 people has 30 options - 30*30*...*30 = 30^20.
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Joined: 02 Sep 2009
Posts: 49948
Re: Twenty people at a meeting were born during the month of Sep  [#permalink]

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13 Aug 2014, 07:42
gigi66653 wrote:
at least two of the people = 1- no two people share the same bday
no two people share the same bday = (1st pick a day in the 30 days) * (2rd pick another day in the left 29 days)
= (1/30) * (29/29)
so, at least two of the people share differ = 1-(1/30) * (29/29) = 29/30 = 99%

It should be 30/30*29/30*28/30*27/30*26/30*...*11/30.
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Twenty people at a meeting were born during the month of Sep  [#permalink]

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02 Nov 2015, 20:48
It is easy to solve the problem by finding the probability where each person is born on a different day and subtracting it from 1.

Let us start with a single way. The first person can be born on Sep 1 , the second on Sep 2 and so on. So the probability of 20 persons born on different days = (1/30)*(1/30) *..20 times =1/(30^20)

How many such ways are there?

(1) the 20 days can be chosen from 30 days in 30C20 ways
(2) The birthdays of the 20 persons can be arranged in 20! ways

For the probability, we have to multiply 1/(30^20) by 30C20 and 20!

So the probability that the birthdays fall on different days = 30C20 * 20! / (30^20)

The probability that at least two persons share the same birthday is 1 - (30C20 *20!) / (30^20) = 99%(approx)
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Re: Twenty people at a meeting were born during the month of Sep  [#permalink]

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20 Sep 2018, 08:01
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