Markov Inequality: A mental model

Markov Inequality is one of the most useful mental model I have ever come across. In my opinion, a good mental model not only helps us to draw good predictions (nice but not the most important thing), but it holds a mirror to our estimations and ideas.

• What is the probability that political party ‘A’ gains above 273 seats in the House of Commons when the expected* number of seats is about 200.
• What is the probability that the Acme Inc will beat the market estimates in topline, when the expected* sales in such economic scenario is about 1/3 of market estimates?

If such questions can be answered with relative confidence (approximately and not necessarily precisely) then as investors, as wise citizens of the world we will be better disposed.

So what does Markov Inequality say?

It is extremely simple (and thus powerful). It says:

If we have a random variable $X$ (what is a random variable? Check here) which is always positive (e.g. the number of seats a political party wins, the sales of a company)  and has an expected value of $E[X]$, then the probability that $X$ exceeds the value of $a$ is less than equal to the ratio of $E[X]$ over $a$.

Mathematically,

$P(X \geq a) \leq \frac{E[x]}{a}$

Caution:

(*) Expected value of X, doesn’t imply psychological expectation, but the average of X over a long period of time.

It is more close to what behavioural psychologists like Kahnemann call as “base rate”.

How to apply it?

For example Acme International has delivered a sales of $150mn , in the past when faced with such macroeconomic conditions. However market is expecting Acme International to post a 20% growth rate over the previous year’s sales of$210mn.  Or in other words, market expects a sales of \$252mn

Thus,

Probability that sales will exceed market expectations $\leq \frac{150mn}{216mn}$

Or less than 60% chance.

Now how to use this information?

We can use this information to ‘shade’ our own estimates, form our own expectations  etc. Also we can form decision trees, make our outcome ‘fans’.