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a space probe near neptune communicates with earth using bit strings. suppose that in its transmissions it sends a 1 one-third of the time and a 0 two-thirds of the time. when a 0 is sent, the probability that it is received correctly is 0.6 and the probability that it is received incorrectly (as a 1) is 0.4. when a 1 is sent the probability that it is received correctly is 0.8 and the probability that it is received incorrectly (as a 0) is 0.2. (a) Find the probability that a 0 is received. (b) Use Bayes theorem to find the probability that a 0 was transmitted, given that a 0 was received

Sagot :

(a) The probability that a 0 is received is 66.7%

(b) By using the Bayes theorem, the probability that a 0 was transmitted, given that a 0 was received is 90%

Probability:

In statistics, probability deals with finding out the likelihood of the occurrence of an event.

Given,

Here the space probe near Neptune communicates with earth using bit strings. suppose that in its transmissions it sends a 1 one-third of the time and a 0 two-thirds of the time. when a 0 is sent, the probability that it is received correctly is 0.6 and the probability that it is received incorrectly (as a 1) is 0.4. when a 1 is sent the probability that it is received correctly is 0.8 and the probability that it is received incorrectly (as a 0) is 0.2.

Now, we have to find the probability of the following.

Here the probability that a 0 is received is calculated as,

=> 0.9 of 2/3(0 received when a 0 is sent) and 0.2 of 1/3(0 received when a 1 is sent). So

Now, the probability is

=> p = (0.9 x 2)/3 + (0.2 x 1)/3

=> p = 0.6667

When we convert this into percentage then we get, 66.67% probability that a 0 is received.

Similarly, by using the Bayes theorem, the probability that a 0 was transmitted, given that a 0 was received is calculated as,

Here we know that, Event A: 0 received and Event B: 0 transmitted.

Then 0.6667 = 66.67% probability that a 0 is received, which means that

P(A) = 66.7%

And the zero is transmitted two-thirds of time, which means that

P(B) = 66.7%

Where as if 0 is sent, then the probability that it is received correctly is 0.9, which means that

P(A|B) =0.9

Therefore, the probability is

=> 66.7 x 0.9/66.7

=> 0.9

We know that 0.9 = 90% probability that a 0 was transmitted, given that a 0 was received.

To know more about Probability here.

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