I can give a quicker explanation.
is the space
Now it is clear that
I understand how the question arrived at the conditional probabilities, however couldn't the question also be interpreted this way?
Since the question never stated that the 2 dice were different, what if we assume they were not distinct and also we can assume that the 2 dice is thrown at the same time, thus
Now to have a sum of 4 and a max of 3, all we need is to roll a 1 and a 3, there is no order here since we are throwing the dice at the same time and the dice are NOT distinct. Thus the numerator becomes 1/6 * 1/6
To get a maximum of 3, all we need is to roll a 1 and a 3 or 2 and a 3 or 3 and a 3 (again there is no order since we are throwing the dice at the same time)
Thus the denominator is 3*(1/6 * 1/6)
Thus ?
Would the above also be a valid interpretation?
Absolutely not. If you roll two dice the outcome space is a set of 36 ordered pairs.
If you roll three dice the outcome space is a set of ordered pairs.
If you roll n dice the outcome space is a set of ordered pairs.
It makes no difference if you roll n dice at once or one die n times the outcome space is the same.
Hi again, I just have an extension to this problem:
I was wondering in order to compute E(S) using the method they described, is it really as tedious as the following? (Obviously I could just apply the definition of straight away, but here I wanted to try their exercise)
So in order to compute E(S) using the method they described we have to work out E(S|M=y) first which is given by
Now we have do that for : which means we have to manually find for
Then we have to manually find for
Then after all that sub back into
I did all of the above and ended up with so much working, is there any shortcuts rather than applying the definitions of each expression directly?