Find the pmf of a dice that tossed

Question

A biased die is tossed twice. Let X be the number of dots faces up in one toss. Below is the pmf of X:

X        1      2    3     4     5     6      otherwise
pX(x)    3/20   1/5  1/10  3/20  1/10  3/10   0

Y be the sum of numbers of dots that face up in two tosses. Find the pmf of Y.

So I think that the pmf of Y should be:

Sum  Prob
2    1/36
3    2/36
4    3/36
...
12   1/36
otherwise 0

Am I right? But what is the point of telling me the pmf of X?

Answer 1

The die is biased, your answer assumes that the die is unbiased.

Compute pmf of $Y$ by

$$P(Y=k) = \sum_{i=1}^6 P(X=i)P(X=k-i)$$