Certainly! Let's carefully determine the probability of each bit being received at the receiver's end, given the probabilities of transmitting each bit and the conditional probabilities of receiving a bit given the transmission.
### Step-by-Step Solution:
1. Define the Probabilities:
- Probability of transmitting a 0 bit: [tex]\( P(T=0) = 0.45 \)[/tex]
- Probability of transmitting a 1 bit: [tex]\( P(T=1) = 0.55 \)[/tex]
- Probability of receiving a 0 bit given that a 0 bit was transmitted: [tex]\( P(R=0 \mid T=0) = 0.8 \)[/tex]
- Probability of receiving a 1 bit given that a 1 bit was transmitted: [tex]\( P(R=1 \mid T=1) = 0.8 \)[/tex]
- Probability of receiving a 1 bit given that a 0 bit was transmitted: [tex]\( P(R=1 \mid T=0) = 0.2 \)[/tex]
- Probability of receiving a 0 bit given that a 1 bit was transmitted: [tex]\( P(R=0 \mid T=1) = 0.2 \)[/tex]
2. Calculate the Probability of Receiving a 0 Bit:
- The probability of receiving a 0 bit ([tex]\( R=0 \)[/tex]) can be found by considering both scenarios where:
- A 0 bit is transmitted and correctly received as a 0 bit.
- A 1 bit is transmitted but incorrectly received as a 0 bit.
- Thus, the formula to find the probability of receiving a 0 bit is:
[tex]\[
P(R=0) = P(T=0) \cdot P(R=0 \mid T=0) + P(T=1) \cdot P(R=0 \mid T=1)
\][/tex]
- Plug in the values:
[tex]\[
P(R=0) = (0.45 \cdot 0.8) + (0.55 \cdot 0.2)
\][/tex]
- After calculating this, you will find:
[tex]\[
P(R=0) = 0.36 + 0.11 = 0.470
\][/tex]
3. Calculate the Probability of Receiving a 1 Bit:
- Similarly, the probability of receiving a 1 bit ([tex]\( R=1 \)[/tex]) can be found by considering:
- A 0 bit is transmitted but incorrectly received as a 1 bit.
- A 1 bit is transmitted and correctly received as a 1 bit.
- The formula to find the probability of receiving a 1 bit is:
[tex]\[
P(R=1) = P(T=0) \cdot P(R=1 \mid T=0) + P(T=1) \cdot P(R=1 \mid T=1)
\][/tex]
- Plug in the values:
[tex]\[
P(R=1) = (0.45 \cdot 0.2) + (0.55 \cdot 0.8)
\][/tex]
- After calculating this, you will achieve:
[tex]\[
P(R=1) = 0.09 + 0.44 = 0.530
\][/tex]
### Final Answer:
- The probability of receiving a 0 bit at the receiver [tex]\( P(R=0) \)[/tex] is [tex]\( 0.47 \)[/tex].
- The probability of receiving a 1 bit at the receiver [tex]\( P(R=1) \)[/tex] is [tex]\( 0.53 \)[/tex].
By understanding these steps, you can identify how the given probabilities for both transmission and reception combine to produce the final probabilities of receiving each bit at the receiver end.
