If I toss two indistinguishable dice, what is the distribution for the sum of the two numbers I get?
Why?
If I toss two indistinguishable dice, what is the distribution for the sum of the two numbers I get?
Why?
The probaility of throwing a total of n with two dice is $\frac{n-1}{36}$ if $n\leq7$, $\frac{13-n}{36}$ if $n>7$ and 0 otherwise. The probability is proportional to the number of ways of expressing n as the sum of 2 numbers in the range 0-6.
With x dice, the proability of a total of n if $x\leq n \leq \lfloor{ \frac{7x}{2}}\rfloor $, if the entry a is in the n-x+1 th row and xth diagonal of Pascal's triangle, is $\frac{a}{6^{x}}$. For $\lceil{ \frac{7x}{2}}\rceil \leq n \leq 6x$, the probability is the same as for $7x-n$.
Mehrdad, Angela is correct. Your professor was screwing with you, possibly in an effort to get you to think for yourself. And please, please, read the book. Carefully. I have rarely let a false idea hold past the end of a class, but some people do it.
If two dice cannot be distinguished, throw them, say 1000 times, taking careful notes of what happens, how many times each total holds.
Now, get some paint, put a tiny green dot on one die, a tiny red dot on the other. Throw them another 1000 times. The proportions of 2,3,4,5,6,7,8,9,10,11,12, are very similar to the first 1000 times. However, it is obviously sensible to draw a square, one row each for the red die, (1-6), one column each for the green die (1-6). In the square you have outlined, put the sum in each box. One may now simply count, there are 36 things that can happen, 6 of them total to 7, so the probability of throwing a 7 is 1/6. The probability of throwing an 8 is 5/36, and so on.
You can construct a table of the probability of throwing a particular pair of numbers like this
Larger 1 2 3 4 5 6 Smaller 1 1/36 2/36 2/36 2/36 2/36 2/36 2 0/36 1/36 2/36 2/36 2/36 2/36 3 0/36 0/36 1/36 2/36 2/36 2/36 4 0/36 0/36 0/36 1/36 2/36 2/36 5 0/36 0/36 0/36 0/36 1/36 2/36 6 0/36 0/36 0/36 0/36 0/36 1/36
Then the probability of throwing a particular total is the sum of the probabilities on a diagonal corresponding to that total. For example the probability of a total of 4 is $\frac{0}{36}+\frac{1}{36}+\frac{2}{36} = \frac{1}{12}$. This will give Angela's results.
I think that the professor uses the term indistinguishable because he may think that a roll of (4,3) is different from a roll of (3,4) when the dice are distinguishable but not so when they are indistinguishable. But it should affect the distribution of the sum of the numbers on the dice.