the birthday problem

[new fish]

hi guys
some of you will be familiar with the birthday problem

"how many ppl do you need in a room before the probability of two sharing a birthday becomes at least 50%"

the answer to this is 23 people

you can reason it by considering the prob that all have different birthdays:
consider a group of 3 ppl
the number of possible combinations of birtdays for the 3 is (366)(366)(366)
now the first person can have a bday on any of 366 days
the second person can have a bday on any of 365 days
and the third person can have a bday on any of 364 days

so the prob that they all have different bdays is
(366*365*364)/(366*366*366)

so for them to share is 1-(the above)

for a group of 23 ppl this yields just over 50%..



NOW
shrink the problem to the chances that in a group of 3 people, 2 were born on the same day of the week, according to the above the probability is
1-(7*6*5/7^3) which is about 38.7%. i have even written a vb simulation to verify all of this and it works out!!
BUT
i cant help thinking
>all 3 choose a day of the week
>person 1 stands up and says his day
the chances that one of the other 2 picked this day are 2/7
>if neither did choose the same day, person 1 leaves the room along with
the day that he choose
>person two stands up and says his day
the chances the third person picked that day are 1/6

so that it seems the chances should be 2/7+1/6 =45.2%
or even at a stretch 2/7 for a first time hit, 5/7 for a fist time miss*1/6 for a second time hit
2/7+(5/7*1/6)

i know this is WRONG but i am trying to think of it i terms of "what are the chances" not "what are the chances of not"

i just thought from my frustration that i'd share it!!!
and see if anyone has way of explaining it through positive outcomes!!