Consider two wireless transmitters, $A$ and $B$, that are near each
other. In each time slot, $A$ transmits a message with probability $1/2$ and $B$ transmits with probability $p$. Whether or not $B$ transmits is independent of whether or not $A$ transmits in the slot. Simultaneous transmissions result in loss of a message. Let $X$ denote the number of time slots until the first message gets through.
(a) Find the pmf of $X$
(b) Given there was a successful transmission, find the probability that $B$ was transmitted.
I interpreted this question as transmission of $B$ occurs only if $B$ passes and $A$ doesn't. This happens with probability $p/2$. Also, $A$ transmission happens when $A$ happens and $B$ doesn't. This happens with probability $(p – 1)/2$. Finally, there is a loss of message if they both transmit. This happens with probability $p/2$.
But then there are three outcomes, so I think I'm interpreting this wrong. I learned about geometric random variables, and I know the PMF of it, but isn't one of the conditions for there to be only two outcomes?
Can someone please help me with this question?
Best Answer
You have successfully (barring one typo) found the distribution of outcomes for a given time slot. Now you have to look at a sequence of independent time slots.
In part (a), you only care about whether something was successfully transmitted or not. You have correctly computed that a successful transmission happens with probability $p/2 + (1-p)/2 = 1/2$. You can also check that therefore the probability of no successful transmission (whether it be because of simultaneous transmission or because of no attempted transmissions) is also $1/2$. These are your two success/failure outcomes and you can now set up your geometric random variable, which you seem to be familiar with.
For part (b), just start with the definition of conditional probability and break it down. $$P(\text{$B$ sent} \mid \text{successful trans.}) = \frac{P(\text{$B$ sent successfully})}{P(\text{successful trans.})}.$$ I computed the denominator in the above paragraph already. You have already computed the numerator in your post.