Solve the probability distribution and expectation problem

[chwala]

Homework Statement

See attached

Relevant Equations

understanding of probability distribution concept...

This is the problem;

Find my working to solution below;

find mark scheme solution below;

I seek any other approach ( shorter way of doing it) will be appreciated...

 

[haruspex]

Homework Statement:: See attached
Relevant Equations:: understanding of probability distribution concept...

shorter way of doing it

It would have been quicker to have found the probability of Y=2 by subtracting the other two probabilities from 1. I don't see any other shortcuts.

 

[chwala]

It would have been quicker to have found the probability of Y=2 by subtracting the other two probabilities from 1. I don't see any other shortcuts.

That's true, but it could pose problem to a student who may have made a mistake on say finding wrong values of $Y=0$& $Y=4$, ...if you get what I mean... this error would consequently affect the value of $Y=2$.
Finding the values of $Y$ indepedently and then checking whether their total sum is $1$ is more concrete...