The detective game involves 6 suspects, 6 weapons and 9 rooms. An item of each set is selected at random and the objective of the game is to find out which of those were choose.
In a version of this game, the itens are selected and then each player recieves, at random, 3 of the remainder cards. Let $S,R,W$ denote the number of suspects, rooms and weapons recieved by an especified player and $X$ is the number of possible soluctions after the player observes his cards.
a) Write $X$ in function of $S, W, R$
b) Find $E(X)$
My attempt
a) Using the fundamental principle of counting, $X=(6-S)(6-W)(9-R)$, right?
b) Unfortunatelly, I think $S,R,W$ are not independent so I couldnt find a good way to use the last item to calculate this one. I noted that each are hipergeometric variables.
I appreciate some help.
Thanks!
You look right about part a. To find part b you can just create a probability mass function for X and use the familiar $\sum xp_X(x)$ to find its expected value.
Finding the probability function isn't really as bad as it might seem since there's some symmetry with $S$ and $R.$ In fact you only have 6 possible values of $X$:
$$X=(6-S)(6-W)(9-R)=\cases{ 162 & for (6-3)(6-0)(9-0) and (6-0)(6-3)(9-0) \\ 180 & for (6-2)(6-1)(9-0) and (6-1)(6-2)(9-0) \\ 192 & for (6-2)(6-0)(9-1) and (6-0)(6-2)(9-1)\\ 200 & for (6-1)(6-1)(9-1) \\210 & for (6-1)(6-0)(9-2) and (6-0)(6-1)(9-2)\\216 & for (6-0)(6-0)(9-3)}$$
And for any $S$, $W$, and $R$, the probability of drawing $s$ $S$ cards, $w$ $W$ cards, and r $R$ cards is
$${5\choose s}{5\choose w}{8\choose r}\over{18\choose 3}$$
so that
$$p_X(x)=\cases{ 0.02451 &x=162 \\ 0.1225 &x=180 \\ 0.1961&x=192\\ 0.2451 &x=200 \\0.3431&x=210\\0.06863&x=216}$$
So $E[X]=(162)(0.02451)+(180)(0.1225)+(192)(0.1961)+(200)(0.2451)+(210)(0.3431)+(216)(0.06863)\approx199.6$