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Three heads up



Chris wrote:
> At 06:21 PM 10/16/2003 -0400, Kent McLean wrote:
> >Look at it this way:
> >
> >Toss 1:    H-H-H
> >Toss 2:    H-H-T
> >Toss 3:    H-T-H
> >Toss 4:    H-T-T
> >Toss 5:    T-T-T
> >Toss 6:    T-T-H
> >Toss 7:    T-H-T
> >Toss 8:    T-H-H
>
> But we are not concerned with the ORDER in which the coins fall as
heads or
> tails.

True, but you can't ignore the outcome just because
the number of heads (or tails) matches another result.
The chances of coming up with two heads is 3/8
(tosses 2, 3, and 8). If you aren't concerned with the
order, then by your argument, the probability for 2
heads is 1/4, the same as for 3 heads, which isn't true.

To prove it, let's toss those three coins 1,000 times.
Each time 3 Heads come up, I'll give you a dollar.
Each time just 2 Heads come up, you'll give me a
dollar.  No one gets paid if the tosses produce 1
Heads or 0 Heads.  Since you are not concerned
with order, I get paid if the coin toss results in
H-H-T, H-T-H, T-H-H. That is, I won't let you
throw out any 2 Heads results, since we are not
concerned with the order. Are you game?
[Hint: try it by tossing 3 coins for just 8 sets.]

Heck, I'm feeling generous.  I'll pay you *$1.10*
for every three Heads. You give me just $1 for
every two Heads. Are you still game?
[Hint: I'm not being generous.]

HTH,
Kent