# If I role two dice, their sum is most likely 7. So if I role a 2 with one die, is the next roll most likely to be a 5?

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Seems weird any way I think about it. Thanks!

In: Mathematics

Well the second die has no statistical relationship to the first die, so you have a 1/6 chance of rolling any number, so no.

No. The reason why 7 is most likely is becsuse no matter what you roll with dice one your second roll still can add up to 7. Which is not the case for other numbers: if you want to roll six, your first roll can only be 1-5 etc..

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if my math is right, you have one third chance to have two dice to add up to 7. This is the highest probability out of all sums from two dice.

The reason that the most likely sum of two 6-sided dice is 7 is that it’s the most common value that comes up when you make a list of all the sums of two numbers between 1 and 6. 7 can result from 1+6, 2+5, 3+4, 4+3, 5+2, and 6 +1, which makes up 6 out of the 36 possible combinations. Every other number in that range has a smaller number of possible combinations. It’s not that the first dice is magically influencing the second, it’s just that statistics says that 7 has to come up the most because it’s got the most entries on the list.

Nope. it’s just that no matter what you roll on the first die, there’s always SOMETHING you can roll on the second die to get a 7, so the chances are 1/6.

If you want to roll a 2, you have to roll a 1 on both rolls, so the chance is 1/36. Same chance for rolling 12.

To roll a 3, you have a 1/3 chance to roll the required 1 or 2, then 1/6 to pick up whichever you missed on the second roll, so the total is 1/18.

The fact says: if you roll two dice, the most probable outcome is 7. This is true, there are simply more combinations of two dice that give 7 than give any other number.

If you are roll a 2 with one dice and then make that result a fact, and then judge the probability of rolling a 5 afterwards, that is no longer the outcome of two dice rolls! That is the outcome of the second dice GIVEN THAT the first dice rolling 2.

In other words you are only most likely to get 7 if you roll the two dice together over and over. If you roll a 2 and then only roll ONE dice after that over and over, there is no one result that is more likely than any others.

Different roles of different or the same die have no connection with each other.

Each new roll has the same chance of outcome as the last. There is no memory involved.

Rolling a 6 two times in a row does not make the next roll anymore or less likely to be a 6.

If you roll a die twice (or two dies once) the possible outcome average out to 7 and make 7 the most likely sum you will get.

If you roll a die once, all six different outcomes are equally likely and they will average out to 3.5.

If you roll two die you will have 36 potential outcomes.

1st die | 2nd die | Sum
—|—|—-
⚀ | ⚀ | 2
⚀| ⚁ | 3
⚀|⚂| 4
⚀|⚃ |5
⚀|⚄ |6
⚀| ⚅|7
⚁| ⚀ | 3
⚁| ⚁ | 4
⚁ |⚂ | 5
⚁ |⚃ |6
⚁ |⚄ |7
⚁ |⚅ |8
⚂| ⚀ | 4
⚂| ⚁ | 5
⚂|⚂ | 6
⚂|⚃ |7
⚂|⚄ |8
⚂| ⚅ |9
⚃ |⚀ | 5
⚃ |⚁ | 6
⚃ |⚂ | 7
⚃ |⚃ |8
⚃ |⚄ |9
⚃ |⚀ |6
⚄ |⚁ | 7
⚄ | ⚂| 8
⚄ |⚃ |9
⚄ |⚄ | 10
⚄ |⚅ | 11
⚅|⚀ | 7
⚅|⚁ | 8
⚅|⚂ | 9
⚅|⚃ | 10
⚅|⚄ | 11
⚅|⚅| 12

As you can see, you have 6 out of 36 chances of having sum of 7, 5 out of 36 chance for having a sum 8 or 6 each, 4 chances each of having 9 and 5 and so on and just one chance for having a 2 or 12.

So 7 is not just the result that everything else averages out to it is also the result that you are most likely to get.

Once you have thrown a die and wait for the result of the 2nd die you can throw 30 of the 36 entries in the table above away. you will still have a 1 in 6 chance of both dies adding up to 7 but there will be 5 other equally likely results that are not 7.

The most likely outcome is 7 of “rolling two dice”.

Having rolled one dice, you have changed the possibilities.

If you want to get 7, there’s always hope after the first roll. No matter what the first roll is, there’s one face you could roll on the second that would add up to 7.

If you’re targeting a number larger than 7, there are some rolls of the first die that will leave you hopeless:

– If you want to get 8, it’s hopeless if the first roll is 1.
– If you want to get 9, it’s hopeless if the first roll is 1-2.
– If you want to get 10, it’s hopeless if the first roll is 1-3.
– If you want to get 11, it’s hopeless if the first roll is 1-4.
– If you want to get 12, it’s hopeless if the first roll is 1-5.

Likewise if you’re targeting a number smaller than 7, you could make your target or overshoot it on the first roll, and the second roll will always add at least 1:

– If you want to get 6, it’s hopeless if the first roll is 6.
– If you want to get 5, it’s hopeless if the first roll is 5-6.
– If you want to get 4, it’s hopeless if the first roll is 4-6.
– If you want to get 3, it’s hopeless if the first roll is 3-6.
– If you want to get 2, it’s hopeless if the first roll is 2-6.

That’s why totals other than 7 are less likely: