Respuesta :
Answer:
There is an 85.73% probability that a 0 was transmitted, given that a 0 was received.
Step-by-step explanation:
This can be formulated as the following problem, by the Bayes Theorem:
What is the probability of B happening, knowing that A has happened.
It can be calculated by the following formula
[tex]P(B/A) = \frac{P(B).P(A/B)}{P(A)}[/tex]
Where P(B) is the probability of B happening, P(A/B) is the probability of A happening knowing that B happened and P(A) is the probability of A happening.
In this problem, we have that:
What is the probability that a 0 was transmitted, given that a 0 was received?
P(B) is the probability that a 0 was transmitted. Suppose that in its transmissions it sends a 1 one-third of the time a 0 two thirds of the time. This means that [tex]P(B) = \frac{2}{3} = 0.667[/tex].
P(A/B) is the probability that a 0 was received, given that a 0 was transmitted. When a 0 is sent, the probability that it is received correctly is 3/5 and the probability that it is received incorrectly (as a 1) is 2/5. So [tex]P(A/B) = \frac{3}{5} = 0.6[/tex].
P(A) is the probability that a 0 was received.
One third of the transmissions are 1. In this case, there is a 1/5 probability that a zero is received.
Two thirds of the transmissions are 2. In this case, there is a 3/5 probability that a zero is received.
So:
[tex]P(A) = 0.333*0.2 + 0.667*0.6 = 0.4668[/tex]
Finally
[tex]P(B/A) = \frac{P(B).P(A/B)}{P(A)} = \frac{0.667*0.6}{0.4668} = 0.8573[/tex]
There is an 85.73% probability that a 0 was transmitted, given that a 0 was received.
