Mean of tosses in equal square A fair die with sides numbered 1,1,2,2,3,3 is tossed until all 3 possible results are received. Let X be the number of times the result 1 was recieved. Calculate E(X). Answer by book: 11:6 I tried to assume the die was tossed k times and calculate P(X=x) using the law of total probability but I keep getting sums that I cannot solve.. thanks in advance

For $i=1,2,3$ let $X_{i}$ denote the number of times that result $i$ was achieved, so that $X=X_{1}$ and the $X_{i}$ have the same distribution.

Let $Y$ denote the number of tosses needed to arrive in the situation that all results are achieved.

Then:$$\mathbb{E}Y=\frac{3}{3}+\frac{3}{2}+\frac{3}{1}=\frac{11}{2}\tag1$$(do you understand why?)

But also: $$Y=X_{1}+X_{2}+X_{3}$$so that: $$\mathbb{E}Y=\mathbb{E}X_{1}+\mathbb{E}X_{2}+\mathbb{E}X_{3}=3\mathbb{E}X\tag2$$

Combining $(1)$ and $(2)$ we find: $$\mathbb{E}X=\frac{11}{6}$$