I am trying to solve Exercise 12 of C3_W2_Assignment:

Exercise 12:
Given a n-sided dice (could be fair or not). You throw it twice and record the sum (there is no dependance between the throws). If you are only given the histogram of the sums can you use it to know which are the probabilities of the dice landing on each side?

My Solution:
For simplicity to demonstration, assume n = 3.
And given the histogram of the sum, I should know value of:
P(sum=2), P(sum=3), P(sum=4), P(sum=5), P(sum=6)

And because there is no dependance between the throws:
P(X=1)=P(Y=1) =: P1
P(X=2)=P(Y=2) =: P2
P(X=3)=P(Y=3) =: P3

And I can have equations such as:
P(sum=2) = P(X=1)P(Y=1) = P1^2
P(sum=3) = P(X=1)P(Y=2) + P(X=2)P(Y=1) = 2* P1 * P2
P(sum=4) = P(X=1)P(Y=3) + P(X=2)P(Y=2) + P(X=3)P(Y=1) = 2* P1 * P3 + P2^2
P1 + P2 + P3 = 1 (sum of all probabilities should be 1)

So, I believe from the equations above I should able to solve the value for P1, P2, P3 (each side probability)

And I am not care about the dice is fair or not in the solution.

So my answer is â€śyes, regardless if the die is fair or notâ€ť, but its actually wrong.

Please help to indication which step or assumption is wrong of my solution, thank you very much for your time!

Hello @yusun
I think itâ€™s the step where you assume independence between the throws is incorrect. When you throw the dice twice and record the sum, the outcome of the second throw should be dependent on the outcome of the first throw.

Hi @Isaak_Kamau
Thanks for replaying, but the question has statement â€ś(there is no dependance between the throws)â€ť, thats the reason I am assume independence between the throws.

I still donâ€™t have access to the course but if the question explicitly states that there is no dependence between the throws your approach and equations should be valid.

I read the no dependence as one literally does not affect the other, not that it denies the conditionality. In a sense you are always throwing the second die but only using itâ€™s result based on the outcome of the first. In this way it would change the answer, at least as I understand it.