Probability~~~ ?...?

Question

Alan and Bill play a game as follows: An ordinary dice is rolled and if a six is obtained then Alan wins and if a one is obtained then Bill wins. If neither a six nor a one is obtained then the dice is rolled again until a decision can be made. What is the probability that Alan wins on

(a) the first roll

The answer is 1/6. Why? If the question ask us to find the probability that Bill wins on the first roll, the probability is 1/6 too?

(b) the second roll,

The answer is 1/9.

I thought of doing something like this:

Since the probability that Alan wins is 1/6, the probability of him losing is 5/6.

So, 5/6 * 1/6 = 5/36... =="

(c) the rth roll.

[No idea.]

What is the probability that Alan wins?

[No idea.]

Answer 1

c) (1/6) (4/6)^(n-1)

Answer 2

(a) The answer is 1/6 because when he rolls the dice, there is 1 way he can roll a six out of a total of 6 numbers he can roll. the probability that bill wins on the first roll is also 1/6.

(b) in order for Alan to win on the second roll, he cannot roll a 6 and Bill cannot roll a 1 (otherwise bill would win on the first roll). also, Alan must roll a 6 on the second roll. therefore, the probability he wins on the second roll is (4/6)(1/6)=4/36=1/9.

Answer 3

Since they each have one number out of 6 possibilities, they each have a 1/6 chance of winning on the first roll.

After that it's a bit more complex.

The first roll has 3 possible outcomes:

A wins (1/6)

B wins (1/6)

no winner (4/6)

So chances for A to win on second roll, first there has to BE a second roll (no winner on first, so 4/6) and then A wins (1/6).

Since it's "and", 4/6 * 1/6 = 4/36 = 1/9.

For A to win on the r-th roll, that scenario has to repeat:

no winner for r-1 rolls, followed by A winning:

2/3 ^ (r-1) * 1/6

Since this is a fair game for the two players,

they have equal chances of winning, which is 1/2 for each of them.

The chance of no winner is negligible.

The chance of there being no winner is (2/3)^n, which goes to 0 as n gets large.

For just 10 rolls, it's 1 in 57 (1.7%) (approximately),

for 25 rolls, it's 1 in 25251,

for 50 rolls, it's 1 in 637621500 (637 million).