My daughter threw a die (D6) six times and got: 6,2,2,6,2,6.
This got us wondering whether having only two distinct numbers come up in six throws was very unlikely or would happen fairly often. How would we go about working out the answer to that question? (short of doing it lots of times and counting!)
On
Here is an alternative solution to the (very good) answer already given here:
The number of ways that you can choose $2$ out of $6$ values is $\dbinom{6}{2}=15$.
Given $2$ values A and B, the number of combinations that you can get in $6$ rolls is:
So you can get $6+15+20+15+6=62$ combinations containing A and B.
And you can get $15\cdot62=930$ combinations containing any $2$ out of $6$ values.
The total number of combinations that you can get in $6$ rolls is simply $6^6=46656$.
Therefore, the probability of having exactly $2$ values in $6$ rolls is $\dfrac{930}{46656}\approx0.0199$.
There are a total of $6^6$ possible outcomes of rolling a die six times. How many of those contain exactly $2$ distinct numbers?
Well, there are ${6\choose 2} = 15$ different pairs of numbers from $1$ to $6$, and for each pair of numbers, there are $2^6$ rolls of dice that contain only those two numbers. However, two of those contain only one of the numbers, so $2^6- 2$ is the number of rolls of two numbers that contains both of those numbers.
All together, this means there are $15\cdot (2^6-2)$ good outcomes out of $6^6$, so the probability of this occuring is $$\frac{15(2^6-2)}{6^6} \approx 1.99331\%$$ so what happened is something that happens rougly every once in $50$ tries. Rare, but not much rarer than rolling a double six with two dice, for example.