Sheldon Ross chapter 2 questions 26

Question

The game of craps is played as follows: A player rolls two dice. If the sum of the dice is either a 2, 3, or 12, the player loses; if the sum is either a 7 or an 11, the player wins. If the outcome is anything else, the player continues to roll the dice until she rolls either the initial outcome or a 7. If the 7 comes first, the player loses, whereas if the initial outcome reoccurs before the 7 appears, the player wins. Compute the probability of a player winning at craps. Hint:Let Ei denote the event that the initial outcome is i and the player wins. The desired probability is  $\sum_{i=2}^{12}$P(Ei). To compute P(Ei), define the events Ei,n to be the event that the initial sum is i and the player wins on the nth roll. Argue that P(Ei) =$\sum_{n=1}^{infinte}$ P(Ei,n).

Answer 1

Nice question...the hint made me solve it...First better to find individual probabilities of sum of cubes roled, this one will be,

2 : (1, 1);

3 : (1, 2),(2, 1);

4 : (1, 3),(2, 2),(3, 1);

5 : (1, 4),(2, 3),(3, 2),(4, 1);

6 : (1, 5),(2, 4),(3, 3),(4, 2),(5, 1);

7 : (1, 6),(2, 5),(3, 4),(4, 3),(5, 2),(6, 1);

8 : (2, 6),(3, 5),(4, 4),(5, 3),(6, 2);

9 : (3, 6),(4, 5),(5, 4),(6, 3);

10 : (4, 6),(5, 5),(6, 4);

11 : (5, 6),(6, 5);

12 : (6, 6)

Now direct chances of winning are 7 and 11 and P(Direct Win) = 8/36 and P(Direct loss) = 4/36 (i.e case 2,3 or 12). Now we need to consider cases (4,5,6,8,9,10) where we keep on playing until we get the same event before a 7.

We know P(4)  = 3/36 . Suppose 'n' be no of rolls made. So if I get 4 in the first roll next (n-2) rolls should be anything other than roll of 7 until we get 4 again.Let i is any of (4,5,6,8,9,10).

P(Ei,n) = P(i)^2 (30 − Pi)^(n−2) /36^n .Now take summation from n = 2 to $infty$. you will get G.P series kind of sum.Ultimately you will get.

P(i)^2 / 36(6 + P(i)) .Now add the probabilities for each case (i.e (4,5,6,8,9,10)) along with P(Direct win).

 

Please pardon my writing, not yet well versed inserting symbols in answers.My answer coming as 0.49 almost.