## At Least One Girl

```The probability of rolling a 7 with a pair of ordinary dice is
1/6, because of the 36 possible (and presumably equally likely)
combinations, six of them yield the sum 7.  Now, suppose the dice
are rolled out of our sight, and some honest person who can see
the results tells us that at least one of the die came up 6.  This
restricts the total number of possible combinations to the following
eleven pairs

(1,6) (2,6) (3,6) (4,6) (5,6) (6,6) (6,1) (6,2) (6,3) (6,4) (6,5)

of which exactly two yield a sum of 7, so we would now assess the
probability of a 7 as 2/11, just slightly better than 1/6.  Of course,
the same analysis would give a probability of 2/11 if our honest
friend had reported "at least one 5" instead of "at least one 6".
In fact, we would arrive at the same probability if he reported
"at least one n" for ANY value of n.  This sometimes strikes people
as paradoxical, because obviously we have "at least one n" for SOME
value of n, so why not just assume this without waiting for our
friend to tell us?  It wold seem that this magically improves our
a_priori odds of rolling a seven from 1/6 to 2/11.

This is the kind of connundrum that often puzzles beginning students
of probability.  It's actually a 6x6 version of the classical 2x2
problem that's usually expressed in terms of boy/girl children.  If we
regard a man's two children as a pair of 2-sided dice, each of which
is equally likely to come up as either a "1" (boy) or a "2" (girl),
then what is the probability of "total = 3"?  Here the obvious answer
is 1/2 (just as the probability of rolling a 7 with two 6-sided dice
is 1/6).

Now we alter the problem by having the father provide us with one of
two specific pieces of (factual) information.  He either tells us "At
least one is a boy" or he tells us "At least one is a girl".  Given one
of these pieces of information, what is the probability of "total = 3"?

Strictly speaking, the problem is underspecified in this form, because
we don't know how the father decides which piece of information to
give us if he has a choice (i.e., if he has one child of each gender).
For example, suppose the father tells us "At least one girl" whenever
possible, and only tells us "At least one boy" if he has no choice.
Then obviously if he tells us "At least one boy" we know the
probability of "total = 3" is zero.

The key point here is the difference between

Condition 1:  There is at least one boy.
Condition 2:  There is at least one girl.
and
Condition 3:  We are told that there is at least one boy.
Condition 4:  We are told that there is at least one girl.

When assessing the conditional probability of "total=3" we are not
given C1 or C2; we are given C3 or C4.  C1 and C2 are each true
exactly 3/4 of the time, and half the time they are both true
simultaneously.  In contrast, C3 and C4 are mutually exclusive and
complementary conditions, but we don't really know what fraction
of the time either of them is true.  Therefore, the problem is
indeterminate.

However, by the "unwritten rule of probability riddles" (the UROPR),
let's assume that all specified choices are to be made randomly from
the available options.  Thus we assume that if the man has one child
of each gender he is equally likely to report "At least one boy" as
to report "At least one girl".  It follows that each of these reports
has a probability of 1/2.  For clarity, let's denote C1 as [boys>=1],
and C3 as [told:boys>=1].  On this basis we can use Bayes' formula
to compute the conditional probability as follows

Pr{ [told:boys>=1] AND total=3 }
Pr{ total=3 | [told:boys>=1] } =  ---------------------------------
Pr{ told:boys>=1 }

The denominator is known to be 1/2, and the numerator is 1/4 because
the total equals 3 exactly half the time, and (by assumption) the
father reports "At least one boy" exactly half of those times.  Thus
the conditional probability of "total=3" given that the father tells
us "At least one boy" is exactly 1/2.

Similarly in the 6x6 case we know that  Pr{total=7} = 1/6, and by the
UROPR we have

Pr{ [told:(n's)>=1] AND total=7} = 1/36

for any n from 1 to 6, so Bayes formula gives the conditional
probability

1/36
Pr{ total=7 | [told:(n's)>=1] }   =   -------  =  1/6
1/6

for any n from 1 to 6.  This in no way conflicts with the fact that

Pr{total=7 | [(n's)>=1] }    =   2/11

because the conditions are different.  Let me try to express this
distinction verbally for the 2x2 case.

Suppose there are 100 fathers in an auditorium, and each is the father
of two children.  Each father is instructed to tell you (truthfully)
if at least one of his children is a boy.  This will apply to about
75 of the fathers.  Now, of those 75 Dads, 2/3 (i.e., 50) have a
daughter, and 1/3 (i.e., 25) have two sons.  Thus, if you want to
guess the gender of their "other" child, the chances are 2/3 that
it is a girl.  (Of course, for the remaining 25 fathers - those
who did not report at least one son - you know immediately they
have two daughters.)

However, suppose instead that all 100 fathers were instructed to tell
you either (a) "At least one of my children is a boy" or (b) "At least
one of my children is a girl".  Based on what each father tells you,
you try to guess the gender of his "other" child.  Strictly speaking
this problem is indeterminate, but if it's also stipulated that fathers
with both a son and a daughter should flip a coin to decide what to
tell you, then the probability that the "other" child is of the
opposite gender is exactly 1/2.  The breakdown is

25 have two sons, and they report at least one son
25 have a son and daughter, and report at least one son

25 have a son and daughter, and report at least one daughter
25 have two daughters, and they report at least one daughter

Thus, regardless of what a particular father reports, you have only a
50% chance of correctly guessing the gender of his "other" child.
```

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