Question

Suppose I have 3 dices. I throw them. I want to get a {1,2,5}.
So I do this to get the probabilities:
\[\frac{1}{6} \cdot \frac{1}{6} \cdot \frac{1}{6} \]
In this case, have I considered other combinations like {2, 1, 5}, {5,2,1}, etc?

Answer 1

You have not considered those other cases. You're assuming you get a 1, 2, and 5 in that order.

Answer 2

If I want to consider, what should I do?

Answer 3

You just have to do one more thing. You can just consider all possible orderings. There happen to be \(3!=6\) orderings, and they all have equal probability. Thus, just multiply your original answer by 6.

Answer 4

Suppose if there were repeated values in side it: {2, 2, 5}. Then in this case, would I still multiply by 3!?

Answer 5

long story short, no. I'll explain in a minute.

Answer 6

ok sure. I will wait for your explanation. Thanks! :)

Answer 7

Hold on, for your first question, are you throwing all three dice at the same time? Or one a a time?

Answer 8

Wouldn't it be the same? They are all iid.

Answer 9

It's actually a little bit different. If we throw them one at a time, we do the process I explained above. Otherwise, you have to do something different.

Answer 10

I think it is all thrown together.

Answer 11

I don't think this is it. Even if they are all thrown together, the dices are i.i.d. random variables on their own.

Answer 12

There's a difference between thrown together and thrown at the same time. You get different information about which dice need to which numbers.

Answer 13

hmm... with your second equation, does it count the combinations of it?

Answer 14

If we were looking at \(\{2, 2, 5\}\) instead, we have to look at a couple different cases. Drawing a tree is a good idea here.

Like above, we have a \(2/6\) chance of getting a 2 or a 5. Then, within this probability, there's a \(2/3\) chance we get a 2, and \(1/3\) chance we get a 5. If we get a 5, we need to find the probability of getting 2 2's, and if we get a 2, we find the probability of getting a 2 and a 5.

Answer 15

Also, you were correct in thinking I was wrong with my other explanation. It doesn't actually matter. I was merely getting confused.

Answer 16

oh I see. Thanks!! :)
And just one thing, how do you type the math equations inline? I used the \[ tags but it goes to the next line.

Answer 17

use \( instead of \[

Answer 18

ohh...haha...thanks for your help and explanation! :)

Answer 19

I should note that doing it the other way I suggested is also a valid way of doing it, but you get \[{3 \over 6} \cdot{2 \over 6}\cdot{1 \over 6}={3! \over 6^3}\]Not what I originally got.

Answer 20

Thanks! :)