Question

Consider an unreliable communication channel that can successfully send a message with probability 1/2, or otherwise, the message is lost with probability 1/2. How many 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? Explain your answer.
Hint: treat this as a Bernoulli process with a probability of success 1/2. The question is equivalent to: how many times do you have to try until you get at least one success?

Answer 1

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:

$P(X=x)=\frac{n!}{r!(n-r)!}p^rq^{n-r}$

We want message is received at least once. This can be written as:

$P(X\geq 1)=1-P(x=0)$

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.

$\frac{63}{64}=1-\frac{n!}{0!(n-0)!}\frac{1}{2}^0\frac{1}{2}^{n-0}$

$\frac{63}{64}=1-\frac{n!}{n!}\frac{1}{2}^{n}$

$\frac{63}{64}=1-\frac{1}{2}^{n}$

$\frac{1}{2}^{n}=1-\frac{63}{64}$

$\frac{1}{2}^{n}=\frac{1}{64}$

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.