Answer:
6 times we need to transmit the message over this unreliable channel so that with probability 63/64.
Step-by-step explanation:
Consider the provided information.
Let x is the number of times massage received.
It is given that the probability of successfully is 1/2.
Thus p = 1/2 and q = 1/2
We want the number of times do we need to transmit the message over this unreliable channel so that with probability 63/64 the message is received at least once.
According to the binomial distribution:
![P(X=x)=\frac{n!}{r!(n-r)!}p^rq^{n-r}](https://tex.z-dn.net/?f=P%28X%3Dx%29%3D%5Cfrac%7Bn%21%7D%7Br%21%28n-r%29%21%7Dp%5Erq%5E%7Bn-r%7D)
We want message is received at least once. This can be written as:
![P(X\geq 1)=1-P(x=0)](https://tex.z-dn.net/?f=P%28X%5Cgeq%201%29%3D1-P%28x%3D0%29)
The probability of at least once is given as 63/64 we need to find the number of times we need to send the massage.
![\frac{63}{64}=1-\frac{n!}{0!(n-0)!}\frac{1}{2}^0\frac{1}{2}^{n-0}](https://tex.z-dn.net/?f=%5Cfrac%7B63%7D%7B64%7D%3D1-%5Cfrac%7Bn%21%7D%7B0%21%28n-0%29%21%7D%5Cfrac%7B1%7D%7B2%7D%5E0%5Cfrac%7B1%7D%7B2%7D%5E%7Bn-0%7D)
![\frac{63}{64}=1-\frac{n!}{n!}\frac{1}{2}^{n}](https://tex.z-dn.net/?f=%5Cfrac%7B63%7D%7B64%7D%3D1-%5Cfrac%7Bn%21%7D%7Bn%21%7D%5Cfrac%7B1%7D%7B2%7D%5E%7Bn%7D)
![\frac{63}{64}=1-\frac{1}{2}^{n}](https://tex.z-dn.net/?f=%5Cfrac%7B63%7D%7B64%7D%3D1-%5Cfrac%7B1%7D%7B2%7D%5E%7Bn%7D)
![\frac{1}{2}^{n}=1-\frac{63}{64}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%5E%7Bn%7D%3D1-%5Cfrac%7B63%7D%7B64%7D)
![\frac{1}{2}^{n}=\frac{1}{64}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%5E%7Bn%7D%3D%5Cfrac%7B1%7D%7B64%7D)
By comparing the value number we find that the value of n should be 6.
Hence, 6 times we need to transmit the message over this unreliable channel so that with probability 63/64.