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?

Approaches to 1.

Step-by-step 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:

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:

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.

step-by-step explanation:

let

we know that

the y-intercept is the y-coordinate of the point when the value of x is equal to zero

so

the y-coordinate of the point a is the y-intercept

the answer part a) is

the value of y-intercept is

part b) find the slope m

we know that

the value of the slope is equal to

substitute the values

therefore

the answer part b) is

the value of the slope is equal to

the general answer is

the slope is

the value of y-intercept is