Question

In: Statistics and Probability

(14%) Calculate for the game of craps the probability of (7%) Winning in the 5th throw....

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

(7%) Winning in the 5th 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 4th significant digit such as say 0.08124.

(7%) Losing in the 5th 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 4th significant digit such as say 0.03146.

GIVE TYPED ANSWER ONLY

Solutions

Expert Solution

Hello

a) 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%

I hope this solves your doubt.

Do give a thumbs up if you find this helpful.


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