Two variants of the Chebyshev's inequality.
Formula parameters
Formula variant
Number 1
Bound (k)
Variance (σ²)
Chance is at most
%

Chebyshev's Theorem Calculator

By Maciej Kowalski, PhD candidate

Welcome traveler, to the Chebyshev's theorem calculator, which makes use of the so-called Chebyshev's inequality to compute the probability that we will diverge from what we expect to obtain. Do not be discouraged, young padawan, by this mathematical formalism combined with a vague description of our problem. We promise you that by the end of your visit to our calculator, you will know the Chebyshev's rule inside out!

Chebyshev's theorem formula

A quick glance at our Chebyshev's theorem calculator and you will see two formulas similar to each other. That is because we can find both of them in the literature (or whatever it is that brought you into our loving arms today), depending on which suits us best. Let's start by breaking down the notation used therein:

  • P denotes the probability of an event (the event specified inside the brackets after the symbol).
  • X is a random variable that describes our random experiment. E.g., it can denote the number of sixes we get in ten dice rolls, and the probability associated with each possible result.
  • E(X) is the expected value of our event (as described informally in the section below).
  • σ² (read: sigma squared) denotes the variance of our event. This parameter describes how much the result of our event can differ if we repeat the experiment several times, and how likely it is to happen. Note that it is traditionally given as a square of some value, σ, which can (and does) sometimes appear in calculations without the square.
  • k is the boundary of the result, i.e., how far we want our result to diverge from the expected.

In general, the first (respectively: the second) formula should be read as follows: the probability of the event X to diverge from the expected value E(X) by at least k (respectively: by at least k * σ) is at most σ² / k² (respectively: 1 / k²).

Chebyshev's rule

Pafnuty Chebyshev was a Russian mathematician whose work can tell us a lot about the universe. "The universe? But I've come here to count some formulas!" Well, admittedly, it's not the time and place to think about how fast the universe expands, but maybe it is the perfect opportunity to see that random events are not that random after all? In fact, many random processes in nature can be approximated by the so-called normal probability distribution, which has an elegant structure and useful properties. But let's not dive into the mathematical mumbo-jumbo yet, and try to understand the idea of our fellow mathematician.

Consider the all-time favorite probability problem - tossing a coin. If we flip a coin a hundred times, how many heads will we get? If we do it once, the probability of getting heads is 0.5, because both results have the same likelihood of happening. So if we toss a coin twice, it makes sense to expect one heads, right? Of course, you can never be sure to obtain just such a result, but in general, if half of the time we should get heads, then, in the end, we should get half as many heads as there were tosses.

You might not realize it yet, but we've just informally defined the so-called expected value of an event. It is usually denoted by E(X), where X is the random variable associated with that event. What the Chebyshev's inequality states is that obtaining a result that is far from that expected value is not likely. Indeed, if we think about getting, say, at most twenty-five heads in the hundred tosses, this doesn't seem too likely. And if we get lower to twenty, fifteen or ten, then the probability should fall along with the number. In other words, we would have to be really unlucky for that to happen. To be specific, we can use the Chebyshev's theorem formula to see that the "really unlucky" translates to at most a 4% chance for twenty-five heads, and as low as 1.56% for ten.

How to use Chebyshev's inequality?

Let's see our Chebyshev's theorem calculator in action and try to follow its line of thinking. Say you wanted to impress a pretty friend at a party and told her that you're a famous magician, Chebyshev the Magnificent. Unfortunately, they seem skeptical and ask for a magic trick. Try to play it cool and make mathematics work in your favor.

Take a deck of cards and say that you'll be pulling out one card, checking its color, and putting it back into the deck. You will repeat this 20 times and state that there will be at least ten clubs among the twenty cards you pull. Quite an impressive magic trick, wouldn't you say? Okay, let's see the calculations behind this.

  1. Our random variable X here denotes the number of cards of clubs that we pull out. Each time we take a card, there's a 1 / 4 chance of it being clubs (since there are four suits and all have the same number of cards).
  2. Since we repeat the experiment twenty times and each time we have a chance of success of one in four, the expected value (i.e., the expected number of clubs) is 20 * 1/4 = 5.
  3. We want to pull at least ten clubs, so we want to diverge from the expected value by 10 - 5 = 5. This is our bound k as denoted in the Chebyshev's theorem formula. (Note that our calculations will take into account all possibilities diverging by at least five from the expected value. It will, therefore, also include obtaining zero clubs.)
  4. In our experiment in each try, there are only two possibilities: a clubs card or a not-clubs card. The big-headed mathematicians call such events binomial probability events. What this means to us is that the variance, σ², is then equal to

σ² = no_of_tries * prob_of_success * prob_of_failure = 20 * 1/4 * 3/4 = 3.75.

  1. According to the first formula in the Chebyshev's theorem calculator, the probability of our event is therefore at most

probability ≤ σ² / k² = 3.75 / 5² = 0.15.

This means that we have a chance of at most 15% for our magic trick to work... Hmm, maybe let's reconsider this. Maybe let's say that it was a mis-slip, and you meant a great mathematician, not a magician. That's just as attractive, right?

Other uses and disadvantages

Let's say that you've made a bet that your favorite basketball team will score between 60 and 80 points in the upcoming match. Everyone expects them to score around 70, so all you want is that they don't score 10 points more or less than that expected value. If we want to calculate the risk of you losing your bet, then this sounds suspiciously similar to what we've done above, but this time we don't want to go too far from some value. Fortunately, this is the exact opposite event, and so our formula will still work after a few adjustments:

P(|X-E(X)| < k) = 1 - P(|X-E(X)| ≥ k) ≥ 1 - (σ² / k²).

Since we don't want to get away from the expected value by more than 10, we want our k to be equal to 11 (since 10 is still okay for us). Now σ² is, in general, a very tricky parameter to count. For the sake of our example, let's say that it is equal to 20. This means that your team has had its ups and downs in the last few matches, and getting 50 points or 90 points is not out of the question. This allows us to count our probability from the above formula:

P(|X-E(X)| < 11) ≥ 1 - (σ² / k²) = 1- (20 / 11²) ≈ 0.83,

which translates to a chance of at least 83%. It seems that your bet was a good one!

Note, however, that both the above and our original formula have a significant disadvantage. Namely, if in the original one the fraction on the right hand side is greater than one, then all the Chebyshev's rule gives us is that our probability is at most 1, or, if we translate this to percentage chance, it is at most 100%. It basically means that our event will happen, or will not. Well, that certainly helps...

Maciej Kowalski, PhD candidate