An Answer Page for Exercise 15

Exercise 15

Re-do the computations from the previous section using View #2 instead of View #1. That is, find the house edge for combined don't pass + free odds bets assuming that any game can end in a tie.

We want to compute the house edge for players who make only don't pass bets and free odds bets.

We imagine that we are playing a series of N craps games.
Since we are assuming View #2,   a push ends the current game.

We begin each game by putting the amount B on the don't pass line. If a point is established, we bet as much as we can in free odds. Whenever a 12 is rolled on the come out, we say that the game has ended in a tie; and, the next roll of the dice begins a new game.

We use the following definitions:

For later use, we note that :

The first thing we shall do is determine the probability of establishing a point in any one game.

Thus,   P( no point )   =   1   -   2 / 3   =   1 / 3

In playing N games, we'd expect to have no point N / 3 times.

In any one game, the point probabilities are:

So, in N games, we would expect

Let's use a table to summarize what we've found so far.

To find the total amount we'd expect to bet in N plays, we add the values in the last column of the above table.

Let   T   =   Total amount of money bet in N plays

The average amount bet in N plays is   T / N   =   B + ( 1 / 18 ) ( 3α + 4β + 5γ )

Since free odds bets do not change the expectation, we can compute the average loss from the expectation:

Putting the above two pieces together, we find:

H   =   house edge   =   average loss / average bet

To finish, we need to find α, β, and γ.

As we've seen before:

Let's use   "FDO yes"   to mean that Full Double Odds are in effect   and   "FDO no"
to mean that Full Double Odds are not in effect.

If full double odds do not apply then   3α + 4β + 5γ   =   6fB + 6fB + 6fB   =   18fB

Substituting this into the above equation for H gives us

If full double odds do apply then

Substituting these values into the formula for H gives us

For case 2b in the following table,   we need   f = 2.
f = 2   and   "FDO yes"     →   H   =   ( 9 / 110 )/ 19   ≈   0.430622%.