## Calculating Survivor Expected Longevity

As covered in my Survivor strategy article, my algorithms seek to optimize the expectancy value of “#weeks survival”— in other words, the Expected Longevity.

The meaning of that figure is easy to explain: it is the average number of weeks you can expect consecutive wins, before losing.

Calculating that figure might be less straightforward, but the following should make the math easy to follow.

Let’s first consider the probability of surviving a certain number of weeks.

Assuming an average 80% probability for each week, it’s easy to derive that:

• You have an 80% chance of passing week 1

• You have 80% x 80% = 64% chance of passing week 2

• You have an (80%)^N chance of passing week N.

A plot of survival probability % vs. #games looks like this: (Note again that the final probability is very close to 0. You’re very unlikely to survive.)

Although these numbers don’t directly get us to calculating average survival time, they get us halfway there. We will use them to find the probability of losing at each time point— not earlier or later. There are two ways to get this probability of stopping: (1) from the incremental changes in the above curve (the “derivative”), or (2) from directly calculating the probability of losing at each point.

Let’s do the second method:

• The probability of stopping at week 1 is clearly 1-80% = 20%. Easy.

• The probability of stopping at week 2 is the probability of succeeding to reach week 2 (which is 80%, to pass week 1), and then losing exactly at week 2 (probability of losing is still 20%). So 80% x 20% = 16%.

• The probability of stopping at week 3 is: getting past week 2 (64% as in the above chart) multiplied by the same 20%: 12.8%.

As mentioned, the result of this calculation gives the same result as the first method I mentioned (calculating incremental change in probability, from week to week).

Let’s look at the probability that you fail exactly at each specific week, as seen from pre-season: Note that the most likely week to fail is week 1. That’s because you are 100% expected to make it to week 1, whereas all later weeks carry the risk that you won’t even make it there to exert your chance of failing.

We’re about to use these to make a weighted average, so let’s double check that these probabilities all add up successively to 100%, by the season end: Note that this cumulative probability of failure is, of course, simply the first survival-probability curve at the top, turned upside down (100% - p). So yes, we can still read from this that you are most likely (approaching 100%) not going to survive to week 17— or even week 4.

Finally, we are ready to calculate the average survival time. We simply use these fail% numbers and combine them with week numbers— in other words taking weighted average of weeks. For week 1 it is (20%)(1), week 2 (16%)(2), week 3 (12.8%)(3)…. Taking the full sum gives the result of 4.5 weeks. 4.5 is also where you find the maximum value of all these products: Again, the sum of all these points is 4.5. And this is the number I optimize each week to provide the most optimal pathways for deciding picks. I explain in my Survivor Strategy article why this is the more rational statistical approach.

Please note the above example is simplified with uniform probabilities each week. Finding the optimal calculation in-season is complicated by having different probabilities for every week. You can also see my example of what the resulting paths look like.