What are the odds of rolling seven, six sided dice one time and getting all the same number?

I just rolled seven dice one time and got all ones in that one roll. I am wondering what the odds are or what is the probability of rolling this type of roll.

Thank you.

  • $\begingroup$ $46655:1$, pretty unlikely. $\endgroup$ – vadim123 Jul 18 '15 at 1:09
  • $\begingroup$ The title is pretty unfortunate. It seems to be asking what the probability is that you would go find seven dice and roll them. $\endgroup$ – Matt Samuel Jul 18 '15 at 1:18
  • $\begingroup$ It is very unlikely, vadim123. That's why I was wondering what the odds were. My husband and I were playing a game called 10,000, which uses 7 dice. In one roll I rolled all 1's. I was pretty excited because this roll won me the game and I was really behind. lol $\endgroup$ – Andrea Jul 18 '15 at 1:38

This may help: the chance of getting a 1 is $\frac{1}{6}$, so the chance of getting seven 1s is $(\frac{1}{6})^7$.

However, we also want to count getting seven 2s, seven 3s, etc. There are six numbers, so now we have a chance of $6\cdot(\frac{1}{6})^7$ which is $(\frac{1}{6})^6$


Think about one die to start. The probability that all the numbers will be the same if you roll one die is (obviously) $100\%$. With two dice, there are $6^2=36$ possibilities, and only six of them are the same number twice, so the probability is $1/6$. With three dice, there are $6^3=216$ possibilities, and again, six ways to get all the same number, so $6/6^3=1/6^2=1/36$ is the probability. If we continue this process, it's not hard to see that with seven dice, the probability of getting all the same number is $1/6^6$.


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