Consider the following simple game:
Use two decks of cards to create a new deck consisting of
any 36 red cards plus any 35 black cards. Shuffle this new
71-card deck and draw a card at random. If the chosen card
is black, you win; if it's red you lose.
Show that the probability of winning this simple game is about the same as that for winning a pass line craps bet.
Let P = Pr( win ) = 35 / ( 35 + 36 ) = 35 / 71 ≈ 0.492957746
In craps, the probability to win a
pass line bet is 244 / 495
≈
0.4929292929....
The two probabilities agree out to 4 decimal places.
How we came up with 35 / 71 :
If you expand 244 / 495 as a
continued fraction
,
the partial quotients are:
< 0, 2, 34, 1, 6 >.
So if you construct the table
( see Fig 1 ) that people usually build for
doing continued fraction work and look at the convergents,
the next to last one is 35 / 71.
|
| Fig 1 |