In researching a probability question, I found my answer in this old Math stackexchange question here: If I roll two fair dice, the probability that I would get at least one 6 would be….
However, the answers posted assume that the dice rolls are independent events so that the probabilities can be multiplied.
I was just curious if it makes a difference in the calculation if: the dice are rolled at the same time, or if they are rolled one after another?
Thanks.
Best Answer
There's no difference between the two procedures (throwing simultaneously or one-by-one) as far as independence is concerned. To see this you might go back to the formal definition of independence.
As we see, it's NOT like independence (statistical independence) implies the product rule, rather the concept of statistical independence is defined by the product rule.
It is easy to check from the joint probability distribution that throwing of two dices are statistically independent.
Assume that the die is fair $($i.e. each of the sides come up with equal probability of $\frac{1}{6})$. If we define $X$ to be the number we get from the $1$st die and $Y$ to be the same from $2$nd die, then $$P(X=i, Y=j)=\frac{1}{36}=\frac{1}{6} \cdot \frac{1}{6} = P(X=i)P(X=j),$$ for $i=1(1)6, ~j=1(1)6$.
We can use this to compute $P(X \in A, ~Y \in B)$ which comes out to be $P(X \in A)P(Y \in B)$.
Lastly, I should (informally) say that, in a discussion of statistics and probability, we only care about statistical independence, which is well-defined. In Philosophy, the notion of independence can be way too complicated for mathematicians to handle, so we don't bother about that!