The two-way table summarizes data on whether students at a certain high school eat regularly in the school cafeteria by grade level. \text{Grade}\ \te

A survey of 4826 randomly selected young adults (aged 19 to 25) asked, "What do you think are the chances you will have much more than a middle-class income at age 30?" The two-way table summarizes the responses.
$\begin{array}{cccc}& \text{ Female }& \text{ Male }& \text{ Total }\\ \text{ Almost no chance }& 96& 98& 194\\ \text{ Some chance but }\text{ probably not }& 426& 286& 712\\ \text{ A 50-50 chance }& 696& 720& 1416\\ \text{ A good chance }& 663& 758& 1421\\ \text{ Almost certain }& 486& 597& 1083\\ \text{ Total }& 2367& 2459& 4826\end{array}$
Choose a survey respondent at random. Define events G: a good chance, M: male, and N: almost no chance. Find $P(C\mid M)$. Interpret this value in context.

Jane chooses a number X at random from the set of numbers
{1, 2, 3, 4}, so that
P(X = k) =
1
4
for k = 1, 2, 3, 4.
She then chooses a number Y at random from the subset of
numbers {X, ...,
4
}; for example, if
X = 3, then Y is chosen at
random from {
3,
4}
.
(i) Find the joint probability distribution of X and Y and display
it in the form of a two-way table.
[5 marks]
(ii) Find the marginal probability distribution of Y , and hence
find E(Y ) and V ar(Y ).
[4 marks]
(iii) Show that Cov(X, Y ) = 5/8.
[4 marks]
(iv) Find the probability distribution of U = X + Y . [7