Probably not the Best

Probabilities are tricky - for me, at least. The fact that I can not use them like other numbers although they are sometimes represented as floating point numbers often trips me up.

Case in point: I'm designing an approach to a packet forwarding problem that includes calculating the probability of a packet making it through a portion of a network. An example that helps describe the solution is in the following diagram:

Where $I$ is the ingress node, $E$ is the egress node and the other nodes are represented by the probability that a given packet will be forwarded ($P_{f}$) to $E$ when received from $I$.

For any given path $I$ to $E$ through a single node, the probability the packet will be delivered ($P(I \Rightarrow E)$) is equal to the probability that the intermediary node will forward the packet (ignoring all other details of the medium).

What I [incorrectly] claimed in my presentation of the solution was: if you use all nodes simultaneously and discard duplicates at $E$, the probability of a packet successfully passing from $I$ to $E$ then becomes $max(N)$. Where $N$ is the set of nodes between $I$ and $E$. Considering the example above that equation results in $P(I \Rightarrow E) = 0.82$. This is better than choosing the node with $P_{f} = 0.21$ but it is not the true probability of a packet getting through. 

This design made several rounds of internal review where this error was not noticed; perhaps there is some solace to be had in knowing I am not the only one who is tripped up by this.

When it finally came time to implement this I was discussing the solution with another engineer and he quickly pointed out that my calculation was incorrect. In fact, my estimation was much lower than the actual likelihood of getting a packet through. The true probability of not getting a packet through is the probability of all intermediate nodes dropping the packet.

$$\prod_{i=0}^{n} (1 - P_{f_{n}})$$

The ultimate effect of my proposed approach is the probability of not having that happen or:

$$1 - (\prod_{i=0}^{n} (1 - P_{f_{n}}))$$

For this specific example that is:

1 - ((1 - 0.76) * (1 - 0.21) * (1 - 0.82) * (1 - 0.33))

or 0.977. A much higher value than my originally proposed 0.82.