What is the probability?
What is the probability?
Out of 5 six-sided dice, what is the probability of landing 1 of one number and 4 of one other number?
On the off chance this is a homework question, I’m going to try to talk through the thinking rather than just answering. Please take into account it’s early morning and I had a late night.
First let’s think about the probability of 4 dice landing on the same number. The important bit here is that we don’t care what that number is, just that all four are the same. That means we can ignore what the first one lands on and just calculate the probability of the other three hitting that same number, which is calculated exactly how you’d expect.
For the fifth die, we want it to land on something different than the number chosen by the first four. Let’s say the first four all landed on 3. Again, we don’t care what it is, as long as it’s not a 3. And again, this equation is exactly what you’d expect if you were asked the probability that a die does not land on 3.
Multiply those together, and you should have your answer.
Note that there is one more complication: we are also interested in cases where the different number is not the last one.
Fortunately, this is also easy to solve, since every other position has exactly the same probability.