The Big Six

For 2-6 pips, the combination of both dice showing 1-3 = 3*3 = 9 ways. Additionally, for 6 pips, there are 6 more ways: 1+4, 4+1, 1+5, 5+1, 2+4, & 4+2. As a result, there are 15 ways to toss for 2-6 pips.

However, from the double stake rule, we have to double the weight of these double faces. In other words, for 2-6 pips, there are 3 possible doubles we need to add to the combination: 1+1, 2+2, & 3+3.

Therefore, the expected loss from 2-6 pips = (-12) $\dfrac{(15+3)}{36}$ = (-12) $\dfrac{18}{36}$ = -6 $

For 7-9 pips, if one die shows 6, the other value varies from 1-3. if one die shows 5, the other value varies from 2-4. Then the rest of the combination is: 4+3, 3+4, & 4+4. After all, there are 15 ways to get 7-9 pips, but again, we have to add 4+4 weight, according to the rule.

Therefore, the expected profit from 7-9 pips = (6) $\dfrac{(15+1)}{36}$ = (6) $\dfrac{16}{36}$ = $\dfrac{8}{3}$ $

Finally, for 10-12 pips, there are 6 possible ways: 4+6, 6+4, 5+5, 5+6, 6+5, 6+6. For 5- and 6-doubles, we will add in the weights for these 2 ways as done above:

Therefore, the expected profit from 10-12 pips = (12) $\dfrac{(6+2)}{36}$ = (12) $\dfrac{8}{36}$ = $\dfrac{8}{3}$ $

Hence, the expected value for this 1-dollar bet = 1 -6 + $\dfrac{8}{3}$ + $\dfrac{8}{3}$ = $\dfrac{1}{3}$ $\approx$ 0.333 $.

In other words, statistically you will lose 2 thirds of a dollar if you decide to gamble in this game.