Thread: Can someone please check this for me?

I am sure I know the answer to this question but it is very important that it is correct. Can someone please double check for me to make sure there are no mistakes?

----------------------------------------------------------------------------------------------------------------------
QUESTION: If three (fair) dice are rolled what is the probability of rolling at most two matching numbers? what is the probability of rolling at least two matching numbers?

ANSWER: total possible outcomes of rolling 3 dice = 6*6*6= 216

two matching pair can be achieved by rolling
[x, x, 1]
[x, x, 2]
[x, x, 3]
[x, x, 4]
[x, x, 5]
[x, x, 6]
where x = any number from 1-6

Notice that at x=1 the first set is [1,1,1] which is more than two matching numbers. likewise at x=2 the second set would be [2, 2, 2] and so on for each number, x. therefore there are 5 possibilities in this case for each number in the set.

six numbers each with five possible outcomes = 30 possible outcomes in this set

Also two matching numbers could be achieved by

[x, 1, x]
[x, 2, x]
[x, 3, x]
[x, 4, x]
[x, 5, x]
[x, 6, x]

or

[1, x, x]
[2, x, x]
[3, x, x]
[4, x, x]
[5, x, x]
[6, x, x]

again each number, x has 5 possible outcomes giving us an additional 30 outcomes for each of these two sets.
30*3 = 90
divide by total possible outcomes
90/216 = .4167

AT LEAST two matching numbers includes the possiblity of rolling triples which can occur in six ways [1, 1, 1], [2, 2, 2], [3, 3, 3] and so on
for a total probability of 6/216

adding this to the previous answer we get 6/216 + 90/216 = 96/216 = .4444
----------------------------------------------------------------------------------------------------------------------

again it is extremely important that these answers and the explanations are 100% correct. if anyone could double check the solution for me it would be greatly appreciated.

Thank you

Hello, bm203!

Your second answer is correct; your first answer is off.

Let me show you a way to avoid making long lists . . .

If three (fair) dice are rolled what is the probability of rolling:

(a) at most two matching numbers?

(b) at least two matching numbers?

There are: . possible outcomes.


(a) At most two matching numbers. .This means 0 matching numbers,
. . .1 matching number (whatever that means?!), or 2 matching numbers.

We do not want 3 matching numbers.
. . There are: ways to have 3 matching numbers.

Hence, there are: . ways to have at most 2 matching numbers.

Therefore: .



(b) At least 2 matching numbers.
. . .This means: 2 matches or 3 matches.

For 2 matches, there are 6 choices of for the value of the Pair.
We want: in some order.
. .

For 3 matches: there are 6 choices for the value of the Triple.
. .


Therefore: .

thank you. i did not mean to frame the question is such a way that would imply zero matches should be counted among the solutions. i will rephrase the first part of the question to say "at least two but not three matching numbers" or perhaps i'll simply make it a three part question.

thanks again.