I have been given a question from a peer to answer which I can't do myself. we wish to calculate the probability that a randomly selected
child from class 2 gets a higher mark than a randomly selected child from class 3 (See the table below to get a better picture). It should end up like this:
Essentially, there are 3 classes of different year level that complete the same test. Class 1 (Random Variable Y1) is the youngest (as reflected by the relatively lower scores out of 10 in the test. p4 and p5 are 0.11 and 0.20 respectively.
I am quite honestly stumped as to how to get to the final answer listed above. I began with noticing that some scores from class 2 will always be lower no matter what the the student from class three gets. For example a score of 5 or less will be lower than any score from any class three student (which has a lowest score of 6) so I know for a start the probability is less than 1-0.41 = 0.39 but how does one go about all the combinations of a score being higher that the other thereon?
You draw numbers from this table, thusly. $$\newcommand{\hide}[1]{\underline{\phantom{#1}}} \Bbb P(Y_2>Y_3 ) ~{~=~\sum_{y=0}^{10}\Bbb P(Y_2>y)\Bbb P(Y_3=y) \qquad ~=~ \sum_{y=0}^{9}\Bbb P(Y_3=y)\sum_{z=y+1}^{10} \Bbb P(Y_2=z) \\ ~=~ {6\cdot0.00 + 0.01\;(0.22+0.12+0.04+0.01) \\ + 0.06\;(\hide{0.12}+\hide{0.04}+\hide{0.01})\\+ 0.19\;(\hide{0.04}+\hide{0.01})\\+ 0.39\;(\hide{0.01})}}$$
Well, I started you off. You can do the rest.