Binomial Probability Question-Conditions Met to Use Binomial Formula?

Question

In a communication system, packets are transmitted from a sender to a receiver. Each packet is received with no error with probability $p$ independently from other packets (with probability $1−p$ the packet is lost). The receiver can decode the message as soon as it receives $k$ packets with no error. Find the probability that the sender sends exactly $n$ packets until the receiver can decode the message successfully.

To solve this problem, I just used the binomial formula. I.e.

$\binom{n}{k}p^n(1-p)^{n-k}$

Is this a correct application of the binomial formula? I tried to solve it using other methods, but was unable to come up with anything that seemed to be appropriate.

Answer 1

No, not quite so.   You want the distribution of the count for iid Bernoulli trials until the $k$ th errorfree packet.   This count follows a negative binomial distribution.

Give an iid errorfree rate of $p$, the probability for an arrangement of exactly $k-1$ errorfree packets among $n-1$ packets and then the $k^{\rm th}$ error free placket exactly on the $n^{\rm th}$ arrival is: ___?

$$\mathsf P(N=n)~=~\binom{n-1}{k-1}p^k(1-p)^{n-k}$$