# Probability of one die being more than either or 2 others

#### Domski74

##### New member
Please can you help with how to calculate this?

Thanks

Dom

#### pka

##### Elite Member
Is this the question: Roll three dice, what is the probability that one of the three is greater than either of the other two?
Please be complete in answering. That will save a lot of time and effort.

#### Domski74

##### New member
Roll 2 dice, then roll another, what is the probability that this third die is more than either of the first 2. Is that saying the same thing? Thanks for the help

#### Jomo

##### Elite Member
Hi. Maybe if you showed us your attempt ( which is required by the guidelines you read) we can figure out what the question really is and also see where you went wrong.

I would start off thinking how one die could not be smaller than the other two.

#### Domski74

##### New member
Ok so I mean greater than either of the other 2 not the combination. I created a grid and found that there are 15/36 possibilities that it would be greater than one of the dice, so was thinking that it would be 30/72 that it would be greater than one or the other?

#### Jomo

##### Elite Member
Ok so I mean greater than either of the other 2 not the combination. I created a grid and found that there are 15/36 possibilities that it would be greater than one of the dice, so was thinking that it would be 30/72 that it would be greater than one or the other?
30/72 is the same as 15/36 !!!!

I still do not understand the question. Is one die special (maybe red and the other two are green) and you want to know if you roll all three dice the probability that the red die is smaller than each green die? Can we please see your work showing us how you got 15/36? Please!?

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#### pka

##### Elite Member
Ok so I mean greater than either of the other 2 not the combination. I created a grid and found that there are 15/36 possibilities that it would be greater than one of the dice, so was thinking that it would be 30/72 that it would be greater than one or the other?
Thanks for the clarification. Rolling three dice given $$\displaystyle 6^3=216$$ outcomes in the form of triples.
Now must consider the numbers, look at this list:
$$\displaystyle \begin{array}{*{20}{c}}{\text{max value}}&{\text{#}}&{\text{total}}\\\hline{6|}&{25}&{75}\\{5|}&{16}&{48}\\{4|}&9&{27}\\{3|}&4&{12}\\{2|}&1&3\end{array}$$
The table tells us that there are seventy-five triples in which six it the greatest value.
That that is: the triple $$\displaystyle (6,1,5)$$ has a die with a value greater than either of the other two.
BUT that happens in three ways. Look at the set of pairs $$\displaystyle \{1,2,3,4,5\}\times\{1,2,3,4,5\}$$ there are twenty-five pairs.
Each of those pairs can go with the six in three ways. (Can you tell why we din't to use the one?)
Note that we counted the triple $$\displaystyle (4,5,4)$$ but not $$\displaystyle (4,4,1)$$ WHY?
Add up the third column. What do you get?
That is, we count the triples having a an unique maximum.
What is the answer to this problem?

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#### Dr.Peterson

##### Elite Member
Roll 2 dice, then roll another, what is the probability that this third die is more than either of the first 2. Is that saying the same thing? Thanks for the help
Ok so I mean greater than either of the other 2 not the combination.
As I understand it, you roll one pair of dice, then want to know the probability that a third die is greater than either of the first pair.

That is, you want

P(third die > max of first pair).​

P(max of pair = 1)*P(third > 1) + P(max of pair = 2)*P(third > 2) + ... + P(max of pair = 6)*P(third > 6).​

You can find the probabilities for the pair by looking at your table of 36 outcomes, and the probabilities for the third are simple. What do you get?

#### pka

##### Elite Member
Roll 2 dice, then roll another, what is the probability that this third die is more than either of the first 2. Is that saying the same thing? Thanks for the help
As I understand it, you roll one pair of dice, then want to know the probability that a third die is greater than either of the first pair.
My previous post was written before you completely clarified the setup.
However, the model is the same. The outcomes are still triples. But now the question is to count the triples in which the third entry is the unique maximum. This table tells us all:
$$\displaystyle \begin{array}{*{20}{c}}{\text{max value}}&{\text{# of triples}}\\\hline{6|}&{25}\\{5|}&{16}\\{4|}&9\\{3|}&4\\{2|}&1\end{array}$$
Now the sum of the second column is $$\displaystyle 55$$.
That means that in rolling a die three times there are $$\displaystyle 55$$ times that the third value is the unique maximum of the three.
So if we roll two red dice and then roll one white die, what is the probability that the value on the white die is greater than the values on either of the red dice?