Please, answer both parts, a and b. thanks. space probe near Neptune communicate

Question

Please, answer both parts, a and b. thanks.

space probe near Neptune communicates with Earth using bit strings. Suppose that in its ansmission it sends a I one-third of the time and 0 two-thirds of the time. When a 0 is sent, the probability that it is received correctly is 0.9, and the probability that it is received as a l is 0.1. When a 1 is sent, the Prob 2. A bility that it is received correctly is 0.8 and the probability that it is received as a 0 is 0.2. (a) Find the probability that a 0 is received. (b) Find the probability that a 0 was transmitted, given that a 0 was received.

Explanation / Answer

Here, we are given that:

P(1 sent) = 1/3 and P( 0 sent ) = 2/3

Also, we are given that:

P( 0 received | 0 sent ) = 0.9 and therefore, P( 1 received | 0 sent ) = 1 - 0.9 = 0.1

P( 1 received | 1 sent ) = 0.8 and therefore, P( 0 received | 1 sent ) = 1 - 0.8 = 0.2

a) Using law of total probability, we get:

P( 0 received ) = P( 0 received | 0 sent )P( 0 sent ) + P( 0 received | 1 sent ) P( 1 sent )

P( 0 received ) = 0.9*(2/3) + 0.2*(1/3 ) = (2/3) = 0.6667

Therefore 0.6667 is the required probability here.

b) Now given that 0 was received, probability that 0 was sent is computed using bayes theorem as:

P( 0 sent | 0 received ) = P( 0 received | 0 sent )P( 0 sent ) / P( 0 received ) = 0.9*(2/3) / 0.6667 = 0.9

Therefore 0.9 is the required probability here.

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