Can someone check if I found the conditional pmf and expectation correctly?

Question

I'm not very good at statistic and is given the problem this problem:

Here's the work that I've done:

I was wondering if I should consider all the dots, 21 dots, for the probability for part (iv) to get 9/21 for P(x|A) or just consider the dots where y<x to get my current answer of 9/9?

Also the same thing for part (v), should the probability be ___/21 or ___/9? Any help would be appreciated. Thanks in advance!

Answer 1

Not quite.   It is $\mathsf P(X{\in}A)$ that equals $\sum_{x\in A}\mathsf P(X{=}x)$.   The conditional probability is then:

$$\begin{align}\mathsf P(X=x\mid X>Y)&=\dfrac{\mathsf P(X=x, Y<x)}{\mathsf P(X>Y)}\\&=\tfrac 19\mathbf 1_{x=1}+\tfrac 29\mathbf 1_{x=2}+\tfrac 39\mathbf 1_{x=3}+\tfrac 29\mathbf 1_{x=4}+\tfrac 19\mathbf 1_{x=5}\end{align}$$

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However that you had the correct probabilities in the calculation of your expected value.

Also note how your graphic indicates that the mass of the triangle centres on $X=3$.