A computer A generated some data that is sent to computers B, C and D in a described manner. Find probabilty of computer B receiving correct data.

Question

A computer $A$ generated some data that can only be either $0$ or $1$. The data was sent over local network to computer $B$, which sends it to computer $C$, which consequently sends it to computer $D$. Taking the noises into consideration, the probability of the data getting transferred from one computer to another unchanged is just $35\%$. If it is known that the data that was generated by computer $A$ got correctly to computer $D$, find the probability that computer $B$ received the correct data.

Any tips on how should I start? What's the idea here and how can I interpret the known fact that computer $D$ received the correct data?

Answer 1

Concatenate the data received by $A$, $B$, $C$, and $D$ (so that, for example, if every computer received a $1$ we would write $1111$).

Suppose, WLOG, that the data generated by $A$ was a $1$.

We are given that $D$ receives the correct data, which in this case must be $1$. Then the cases in which $B$ receives the correct data are exactly:

$$1101 \,\,\,\,\,\, \text{and} \,\,\,\,\,\,\, 1111$$

The total sample space is all of the cases in which $D$ receives the correct data, which are exactly:

$$1001 \,\,\,\,\,\, \text{and} \,\,\,\,\,\,\, 1011\,\,\,\,\,\, \text{and} \,\,\,\,\,\,1101\,\,\,\,\,\,\, \text{and} \,\,\,\,\,\,\,1111$$

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The solution to your problem is therefore

$$\boxed{\frac{P(1101) + P(1111)}{P(1001) + P(1011) + P(1101)+P(1111)}}$$

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Each term in the expression above can be computed straightforwardly given that the probability of correct data transfer at any individual step is $35\%$.

For example,

\begin{align*} P(1101) &= P(B=1 \, |\, A = 1)\,P(C=0 \, | \, B=1)\, P(D=1 \,| C=0)\\ P(1101) &= 0.35 \cdot (1-0.35) \cdot (1-0.35) \end{align*}