Pr( N=2 randomly selected persons has DIFFERENT birth MONTHS) -- the definitive answers

Jack Tomsky did a really nice piece if detective work for the
LAST WORD on the correct solutions!!

< snip all nonessentials >

Jack Tomsky wrote:
I got yet a third result for N = 2. I get p =
0.916608. This is calculated from

p = 1-Sum[(Di/365.25)^2],

where Di = 31, 28.25, 31, ..., are the days in each month.

Jack

That turned out to be the correct answer for the probability
of N = 2 people having DIFFERENT birth MONTHS, using
Afonso's assumption that the probability of birth in any given
month is exactly proportional to the number of days in the month.
(a dubious assumption at best that makes the solution of
the problem extremely tedious for N = 3, 4, 5, ..., 12).


Meanwhile, after a late-night filet mignon and having turned
down a $500 bottle of Chateau LaFite Rothschild :-), on
April 7, I took about 2 minutes on the spread*** to know
that Luis Afonso was wrong in the numbers he gave without
showing any formula, because the OBVIOUS formula was
the one I used for N = 2:

RF > The probability of two persons picked at random to
RF > have DIFFERENT birth months is:

RF > Pr(1st person born on the ith month) x
RF > Pr(2nd person born on a different month)
RF >
RF > summed over all 12 months, which gives below,
RF > < the 1st person born on Jan, Feb, ..., Dec, and
RF > > the 2nd person born on a different month>
RF > respectively

and the numbers below were those I got in those two
minutes after dinner. :-)

0.077669885
0.071940262
0.077669885
0.075389279
0.077669885
0.075389279
0.077669885
0.077669885
0.075389279
0.077669885
0.075389279
0.077669885

0.917186573 = Sum of the above,

Because the formula I used was drastically different
(in looks) from the one Jack used, I said to Jack

you had the right
idea and just juxtaposed your "Sum" and "1-Di"
RF > because the prob
RF > of 2 people NOT born on the same month is:
RF >
RF > Sum [ (Di/365.25) x (1-Di/365.25)]

which WAS, and IS, the correct formula!

I also noted,

RF > The probability for N = 2, using the approximation
RF > of the same p for each month would have produced
RF > 0.916666667 for DIFFERENT birth months for N=2

< snipped Jack's (correct) derivation of HIS formula for N=2>

JT > I used the SUMSQ function in Excel to arrive at 0.916608.

JT > Bob, we're using equivalent formulas, so I'm not sure
JT > why we're getting slightly different numerical results.

In a THIRD post, Jack did the detective work, based on the
numbers I gave and found the source of the discrepancy --
MY BAD for the ONE typo of "28.5" for "28.25"!

Bob, I found the cause of the numerical discrepancy. I tried to
recreate the (Di/365.25)(1-Di/365.25) terms which you displayed.
Every month matches up, except February.

JT> You had to have used D2 = 28.5 rather than 28.25 in order to
JT> obtain 0.071940262. With the correct 28.25, you would get
JT> 0.071362146. Then it adds up to my number.

Indeed! Sherlock Holmes would have said to you, "Elementary,
Jack, but very well done!"

Using different prob for the months, the probability of DIFFERENT
birth MONTHS for N = 2 randomly selected persons
= 0.91660846

Using the much simpler approx. of p=1/12 for every month, the
corresponding probability for N = 2 randomly selected persons
= 0.91666666 or 1/12.

The absolute difference between those two results is .00006.


As for the resident ignoramus, Luis A. Afonso, here is his
corresponding number for (the simplest case) N = 2:

Date: Fri, Apr 7 2006 11:43 am

LAA> The problem solved by the Bob´s TWELVES formula
LAA> (which is a rather good approximation, evidently, mine
LAA> is only a refinement):

LAA> p0= (12/12) (11/12) (10/12)...((12-N+1)/12)

LAA> for N=2 gives p0= 11/12 = 0.9167

LAA> When we use my formula we obtain (I checked myself) p=0.9151.

p = 0.9151 (according to Afonso) when the correct answer is .91660846,
which rounded to .9167, the same rounded value as my
much simpler approximate solution!

But that's not all!

LAA> I spent all this night awake, and I am tired in consequence. I
will
LAA> not go further; I will try to get a sleep.

LAA's attempted sarcasm turned out to be a joke on himself, because
he apparently didn't get any sleep because less than 5 hours later:

Date: Fri, Apr 7 2006 4:27 pm
Email: "\"Luis A. Afonso\"" <lic...@xxxxxxxxxxx>

_ N = number of persons
_____prob. no matches
__2__.9151 <============ same mistake
__3__.7667
__4__.5772
__5__.3872
__6__.2269
__7__.1143
__8__.0479
__9__.0160
_10__.0040
_11__.0007
_12__.0001


The absolute error in Afonso's N = 2 case is .9166-.9151 or .0015.

The absolute difference between the exact (.916608457) and the
approximate (.91666666667) results is LESS THAN .00006.

Of course Afonso's other results for N = 3, 4, ..., are all wrong.

We can ALL laugh in unison now, while Luis A. Afonso continued
to do his Mouth Dance with more and more NEW FEET
dangling from his mouth.

-- Bob.

.

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