Question
Roll two six-sided fair dice.
(a) What is the probability that you roll exactly one even and one odd value?
(b) Suppose we roll the dice $n$ times. Let $X=$ the total number of rolls with one odd and one even value. What is the probability distribution function of $X$?
(c) Suppose you win one dollar when you roll one even and one odd value. You also lose one dollar when you roll two odds or two evens. Let $Y=$ the total gain for $n$ rolls. What is $E[Y]$?
I am a little stuck on parts (b) and (c), but hopefully my work isn't too far off!
My Attempt
(a) P(first odd and second even) + P(first even and second odd) = $\frac{3}{6}*\frac{3}{6} + \frac{3}{6}*\frac{3}{6} = \frac{1}{2}$
(b) I'm not positive on the random variable $X$. I have assumed it as $X=0$ where there are no evens. $X=1$ when there is one odd and one even, $X=2$ when there are two evens.
The probability distribution is as follows: $P(X=0)=\frac{1}{4}$, $P(X=1)=\frac{1}{2}$, $P(X=2)=\frac{1}{4}$.
(c) $E[Y] = \frac{1}{4}*(-1) + \frac{1}{2}*(1) + \frac{1}{4}*(-1)=0$. So the total gain for $n$ rolls is zero dollars.
(a) Okay.
(b) You roll the pair of dice $n$ times. That is two dice on each trial. A success on a trial is an odd and an even, occurring with the probability you calculated for $(a)$.
So, for some number $n$, you have $n$ Bernoulli trials with success rate $1/2$. $X$ is the count of successes in these $n$ trials.
(c) As above. Express $Y$ in terms of $X$. Find $\mathsf E(X)$. Then find $\mathsf E(Y)$ from that.
Tip: $Y$ is ...