Respuesta :
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:
[tex]P(X=x)=\frac{n!}{r!(n-r)!}p^rq^{n-r}[/tex]
We want message is received at least once. This can be written as:
[tex]P(X\geq 1)=1-P(x=0)[/tex]
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.
[tex]\frac{63}{64}=1-\frac{n!}{0!(n-0)!}\frac{1}{2}^0\frac{1}{2}^{n-0}[/tex]
[tex]\frac{63}{64}=1-\frac{n!}{n!}\frac{1}{2}^{n}[/tex]
[tex]\frac{63}{64}=1-\frac{1}{2}^{n}[/tex]
[tex]\frac{1}{2}^{n}=1-\frac{63}{64}[/tex]
[tex]\frac{1}{2}^{n}=\frac{1}{64}[/tex]
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.
