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# Probability

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There are 18 people in a team. 7 share the same month of birth(not the exact date). What is the probability of 7 out of 18 having the same birth month?

Mar 8, 2022

#1
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12*nCr(18,7)*(1/12)^7*(11/12)^11 = 0.0040925417608967

That is the probability of getting 7 in the same month but it includes the probability of getting 7 in 2 different months as well.  I mean it includes it twice.

So you would have to subtract

18C7 * (1/12)^7 * (11/12)^11    *    11C7 * (1/12)^7 * (11/12)^4 *  12C2

nCr(18,7)*(1/12)^7*(11/12)^11*nCr(11,7)*(1/12)^7*(11/12)^4*nCr(12,2)=0.0000001463682172

I'm reasonably confident on the first part but have less confidence on the secontd part (the part to be subrtracted)

0.0040925417608967-0.0000001463682172 = 0.0040923953926795

Mar 8, 2022
#2
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Hi Melody:

What do you think of this approach:

First person ==Any month ==12/12

2nd person different from the 1st ==11/12

3rd person different from the first two ==10/12

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And so on to the 12th person ==1/12

So, the probability that the first 12 persons have a different month each is: 12! / 12^12 ==0.0000537232 1709

Or: The probability that at least 2 share the same month is: 1  -  0.0000537232 1709==0.9999462768 - almost a certainty!! Now, I'm stuck on the remaining 6 !!. If the first 12 have each a different month, then the 13th will have 100% chance of sharing his/her month with one of the first 12!! What about the 14th person, and the 15th, 16th, 17th and 18th??

Mar 8, 2022
#5
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It is good that you had a go guest but  I do not think your method can work.

For instance, the second person may not be different from the first,

Melody  Mar 10, 2022
#3
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(18 nCr 7 * 11^11) + (18 nCr 8*11^10) + (18 nCr 9*11^9) +(18 nCr 10*11^8) +(18 nCr 11*11^7) + (18 nCr 12*11^6) + (18 nCr 13 * 11^5) + (18 nCr 14 * 11^4) + (18 nCr 15 * 11^3) + (18 nCr 16 * 11^2) + (18 nCr 17 * 11^1) + (18 nCr 18 * 11^0)==

(907975 8605524464, 113496 9825690558, 11464 3416736420, 937 9915914798, 62 0159729904, 3 2887258404, 1379884968, 44801460, 1086096, 18513, 198, 1, 0, 0, 0, 0, 0, 0, 0)==1033940 6236645784 /12^18== 0.000388358 - probability of at least 7 out of 18 people.

(18 nCr 7) * (11^11)==9079758605524464 / 12^18==  0.000341045 - probability of exactly 7 out of 18 people.

Mar 9, 2022
#4
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0.000388358. That is the at least probability for one specific month, such as January. It should be multiplied by 12 to get: 0.004660296

The same applies to exact probability:  0.000341045  x 12 ==0.00409254 [which agrees exactly with Melody's answer!]

Guest Mar 9, 2022