Question

19. A game is played by rolling a group of at least 3 dice. The highest two numeric results are removed

from the table. If any 6s remain on the table the player wins. If only 3 dice are rolled then the
probability of winning is 1/216. As the number of dice used increases without bound what does the
probability of a winning roll approach?

Answer 1

Answer: Approaches to 1.

Explanation:

If there are only 3 dice used, then the only chance that the player has to win is when the 3 dice have the same outcome, 6.

The probability will be:

p = (1/6)^3 = 1/216.

Now, if we add one more dice, we still need at least 3 sixes to win, but the other dice can have any other value. so now the probabilities are:

dice 1---- outcome = 6, prob = 1/6.

dice 2---- outcome = 6, prob = 1/6.

dice 3---- outcome = 6, prob = 1/6.

dice 4---- outcome = any number, prob = 1.

The probability for this arrangement is still:

p = 1/216.

But now we have permutations!.

The dice that can be any number has 4 possible positions, so the actual probability will be:

P = 4*p = 4/216.

Now remember that if we have N elements, the total number of combinations of K elements ( N ≥ K) is:

`C(N, K) = (N!)/((N - K)!K!)`

if we add other dice, then we will have 5 dices, and 2 of them that can not be 6 that can take any position, then the number of combinations will be:

`C(5, 2) = (5!)/((5 - 2)!2!) = (5*4)/(2) = 10`

Then the probability will be:

P = 10*p = 10/216.

So we can start to see a pattern here, if we have N dices, we still only need 3 of them to be strictly 6, then we have (N - 3) dices that can be any number.

Then the probabilty of winning if you have N dices is:

P = C(N, N - 3)*p = C(N, N - 3)*(1/216)

Then as N increases, we will see that the probability tends to 1, (it actually grows larger than that, but remember that the probability is a number between 0 and 1, so the maximum is 1)

Why? well... if you roll a lot of dice, suppose 1000 of them, is really likely to have at least 3 sixes in there, so as the number of dice increases, also does the probability.