Question

(14%) Calculate for the game of craps the probability of

(7%) Winning in the 5^th throw. Show clearly how you derive the formula; the formula itself; the fractions p/q such as say 125/216, and also the decimal expression to 4^th significant digit such as say 0.08124.

(7%) Losing in the 5^th throw. Show clearly how you derive the formula; the formula; the fractions p/q such as say 73/216, and also the decimal expression to 4^th significant digit such as say 0.03146.

Answer 1

Solution:

To win in the 5th throw, the player must :

(i) Throw a 4,5,6,8,9,10 in the first throw.

(ii) Must not throw the sum in point (i) or 7 in 2nd, 3rd and 4th throw.

(iii) Must throw a sum in first point in 5th throw.

So,

When the initial sum of throws is 4 : P(Winning on 5th throw) = (3/36)*(27/36)*(27/36)*(27/36)*(3/36) = (3/1024) = 0.0029.

When the initial sum of throws is 5 : P(Winning on 5th throw) = (4/36)*(26/36)*(26/36)*(26/36)*(4/36) = (2197/472392) = 0.0047

When the initial sum of throws is 6 : P(Winning on 5th throw) = (5/36)*(25/36)*(25/36)*(25/36)*(5/36) = (390625/60466176) = 0.0065

When the initial sum of throws is 8 : P(Winning on 5th throw) = (5/36)*(25/36)*(25/36)*(25/36)*(5/36) = (390625/60466176) = 0.0065

When the initial sum of throws is 9 : P(Winning on 5th throw) = (4/36)*(26/36)*(26/36)*(26/36)*(4/36) = (2197/472392) = 0.0047

When the initial sum of throws is 10 : P(Winning on 5th throw) = (3/36)*(27/36)*(27/36)*(27/36)*(3/36) = (3/1024) = 0.0029.

Hence, the probability of winning in 5th throw = 0.0281, i.e. 2.81%

b) Losing in the 5th throw, the player must :

(i) Throw a 4,5,6,8,9,10 in the first throw.

(ii) Must not throw the sum in point (i) or 7 in 2nd, 3rd and 4th throw.

(iii) Must throw a 7 in 5th throw.

So,

When the initial sum of throws is 4 : P(Losing on 5th throw) = (3/36)*(27/36)*(27/36)*(27/36)*(6/36) = (3/512) = 0.0059

When the initial sum of throws is 5 : P(Winning on 5th throw) = (4/36)*(26/36)*(26/36)*(26/36)*(6/36) = (6591/944784) = 0.007

When the initial sum of throws is 6 : P(Winning on 5th throw) = (5/36)*(25/36)*(25/36)*(25/36)*(6/36) = (468750/60466176) = 0.0078

When the initial sum of throws is 8 : P(Winning on 5th throw) = (5/36)*(25/36)*(25/36)*(25/36)*(6/36) = (468750/60466176) = 0.0078

When the initial sum of throws is 9 : P(Winning on 5th throw) = (4/36)*(26/36)*(26/36)*(26/36)*(6/36) = (6591/944784) = 0.007

When the initial sum of throws is 10 : P(Winning on 5th throw) = (3/36)*(27/36)*(27/36)*(27/36)*(6/36) = (3/512) = 0.0059.

Hence, the probability of winning in 5th throw = 0.0412, i.e. 4.12%