Question

Three persons, A, B, and C, take turns in throwing a die. They throw in the order A, B, C, A, B, C, A, B, etc., until someone wins. A wins by throwing a "one". B wins by throwing a "one" or a "two". C wins by throwing a "one", a "two", or a "three". Find the probability that each of the players is the winner.

Answer 1

Probability of A winning on a single chance: 1/6

Probability of B winning on a single chance: 2/6

Probability of C winning on a single chance: 3/6

Probability of A losing on a single chance: 5/6

Probability of B losing on a single chance: 4/6

Probability of C losing on a single chance: 3/6

The probability that A wins the game:

We know that A starts the game. Thus probability of winning the game in the first chance is 1/6.

We know that A starts the game. Thus probability of winning the game in the fourth chance is 5/6*4/6*3/6*1/6.

We know that A starts the game. Thus probability of winning the game in the seventh chance is 5/6*4/6*3/6*5/6*4/6*3/6*1/6.

Thus, this keeps going on till infinite.

The total probability= 1/6+5/6*4/6*3/6*1/6+ 5/6*4/6*3/6*5/6*4/6*3/6*1/6+....

This is a geometric progression. a=1/6, and ratio is 5/6*4/6*3/6

Thus, the sum of this GP is a/(1-r)= 1/6 / 1-5/6*4/6*3/6

= 1/6 / 1-60/216

= 1/6 / 156/216

= 1*216 / 6*156

= 216/ 936

= 0.2307692

The probability that B wins the game:

We know that B goes second in the game. Thus probability of winning the game in the second chance is 5/6*2/6.

We know that B goes second in the game. Thus probability of winning the game in the fifth chance is 5/6*4/6*3/6*5/6*2/6.

We know that B goes second in the game. Thus probability of winning the game in the eighth chance is 5/6*4/6*3/6*5/6*4/6*3/6*5/6*2/6.

Thus, this keeps going on till infinite.

The total probability= 5/6*2/6+5/6*4/6*3/6*5/6*2/6+5/6*4/6*3/6*5/6*4/6*3/6*5/6*2/6....

This is a geometric progression. a=5/6*2/6, and ratio is 5/6*4/6*3/6

Thus, the sum of this GP is a/(1-r)= 5/6*2/6 / 1-5/6*4/6*3/6

= 10/36 / 1-60/216

= 10/36 / 156/216

= 10*216 / 36*156

= 2160/ 5616

= 0.3846154

The probability that C wins the game:

We know that C goes third in the game. Thus probability of winning the game in the third chance is 5/6*4/6*3/6.

We know that C goes third in the game. Thus probability of winning the game in the sixth chance is 5/6*4/6*3/6*5/6*4/6*3/6.

We know that C goes third in the game. Thus probability of winning the game in the ninth chance is 5/6*4/6*3/6*5/6*4/6*3/6*5/6*4/6*3/6.

Thus, this keeps going on till infinite.

The total probability= 1/6+5/6*4/6*3/6*1/6+ 5/6*4/6*3/6*5/6*4/6*3/6*1/6+....

This is a geometric progression. a=5/6*4/6*3/6, and ratio is 5/6*4/6*3/6

Thus, the sum of this GP is a/(1-r)= 1/6 / 1-5/6*4/6*3/6

= 60/216 / 1-60/216

= 60/216 / 156/216

= 60*216 / 216*156

= 60/156

= 0.3846154

Thus,

A probability= 23.08%

B probability= 38.46%

C probability= 38.46%