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Probability & Statistics/Probability with 12 sided dice

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Question
Hi,

I would like to know what the probability is of rolling at least one 10, 11, or 12 on two twelve-sided dice.

And then, does the probability of doing this increase when rolling 3, 4, or 5 dice?

Basically, I want to know if it becomes easier to roll a high number the more dice you roll?

Thanks so much for your time,

Tim

Answer
It is easier to compute the probability of two events happening than one and/or the other.

If you are rolling two dice, hoping to get 10,11,12 on either one, it's easier to consider the possibility that you lose (getting neither of them as 10,11,12).

There is a 3/12 chance = 1/4 chance that you get what you want on each die. This means there is a 3/4 chance that you do not.

So for you to fail, you need to make the 3/4 chance of losing on each die.

If you have two dice, the chances of failing are (3/4)^2.

If you have three dice, the chances of failing are (3/4)^3.

Etc.

The chances of losing go down, which means the chances of winning go up. After all, with more dice, it's like you have extra chances to get that 10,11,12 roll.



This chances of winning for each of these is 1-(3/4)^2, 1-(3/4)^3, etc.

The chances of winning with n dice are:

n=2, 7/16 ≈ 43%
n=3, 37/64 ≈ 57%
n=4, 175/256 ≈ 68%
n=5, 781/1024 ≈ 76%

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Clyde Oliver

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I can answer all questions up to, and including, graduate level mathematics. I do not have expertise in statistics (I can answer questions about the mathematical foundations of statistics). I am very much proficient in probability. I am not inclined to answer questions that appear to be homework, nor questions that are not meaningful or advanced in any way.

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I am a PhD educated mathematician working in research at a major university.

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