Probablity of three-of-a-kind or better in a roll of four dice So. I took a math competition and one of the questions seemed simple enough.
"Four fair six-sided dice are rolled. What is the probability that at least three of the four dice show the same value"
Hm. Easy.
Tried solving it.
I couldn't get it.
How can I solve this?
I know you can probably use Combination, and if so, can you step by step explain how (never learned combinations yet).
Or if it's possible without combination, can you explain how to solve it logically? thanks.

We want at least three of the four dice to have the same value. So there are two favourable conditions. 


*

*All the dice have the same value. Let this be event $ A $.

*Three out of the four dice have the same value. Let this be event $ B $.


Let us calculate the probabilities for each of these events. 
For event $ A $, let's say the first die gets a number $ x $. This could be any number from $ \{1,2,3,4,5,6\} $, we don't care. The probability for this to happen is $ 1 $. Now, we want the rest of the three dice to have exactly the same number. The probability for this to happen is $ 1 \times \dfrac{1}{6} \times \dfrac{1}{6} \times \dfrac{1}{6} = \dfrac{1}{216}$.
Fine, now let's calculuate the probability for event $ B$. One of the dice can have any number $ x $. Probability for this to happen is $ 1 $. Now, two out of the three dice should have this same number, and the third dice shouldn't. The probability for this to happen is $ 1 \times \dfrac{1}{6} \times \dfrac{1}{6} \times \dfrac{5}{6} $. Now, remember that we considered this for only one tuple, say dice $ (1,2,3) $ gets the same number. We want this for all tuples. There are a total of $ \dbinom{4}{3} $ tuples. Thus, the total probability for event $ B $ is 
$$  \dbinom{4}{3} \times 1 \times \dfrac{1}{6} \times \dfrac{1}{6} \times \dfrac{5}{6} = \dfrac{20}{216}$$
Thus, our final answer is the sum of the probabilities of events $ A $ and $B$. We get $ \dfrac{1}{216} + \dfrac{20}{216} = \boxed{\dfrac{21}{216}} \approx 9.72\%$.
