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On Wikipedia it says

[T]he binomial distribution with parameters $n$ and $p$ is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own boolean-valued outcome: a random variable containing a single bit of information: success/yes/true/one (with probability $p$) or failure/no/false/zero (with probability $q = 1 − p$)

My own data (viewable here) deals with situations in which a favorite is playing an underdog in sports. I'm interested in whether a strategy of betting on the underdog should lose more money than a strategy on betting on the favorite, as theory would suggest.

In this context I was encouraged in this comment to another question to model either the favorite or underdog data as binomial.

Let's say I pick the favorite to model. Looking through my own data, I have 1434 trials in total, and so I guess for me $n=1434$. Then I worked out that in those games the favorite won the game 1032 times, so for me $p = 0.72$.

However, I'm unsure on some deeper level what it means to model data as binomial. This is going to seem a hopelessly naive question, but I'm having trouble getting my intuition around this concept. In practical terms (either in general, or with reference to my particular situation) what does it mean to model data "as binomial", and how does it help?

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If you model data "as binomial" you assume that:

  1. The probability remains constant over all $n$ trials and

  2. The outcomes of the events are indepdendent from each other

In my examples I will set $n=2$ for simplicity.

Now in order to understand what it means to model something as binomial, you might want to imagine possible violisations of these two assumptions. A possible violisation of (1) could e.g. be if the difference in quality between the two teams varies considerably between the various trials. Imagine for example if the first match was Real Madrid vs Dortmundand the second match was France vs Faroe Islands. The probabilities would be considerably different: While Dortmund has a non-negligible chance of beating Real Madrid, Faroe Islands will probably never beat France So in this case it would be inadequate to model these two games as binomial.

For a violation of the second assumption think of a situation in which the outcome of the first match influences the second match. E.g. imagine that the two matches are Real Madrid vs Villareal and Barcelona vs. Athletic Bilbao. Assume that it is the last game in the season and Real Madrid is first in the leauge with 80 points and Barcelona has 79 points. Furthermore assume that Barcelona plays Athletic Bilbao first. Now if Barcelona loses its match, we could argue that the probability of Real Madrid winning against Villareal is lower, compared to a situation in which Barelona wins and Real Madrid is in a must win situation.

Now if $X_1, ..., X_n$ are the Random Variables that indicate whether the favourite won or the underdog won, the assumption (1) corresponds mathematically with: $$P(X_1=0)=...=P(X_n=0)$$

The assumtion of independence (2) corresponds mathematically with:

$$P(X_1=x_1, ..., X_n=x_n)=\Pi_{i=1}^nP(X_i=x_i)$$ i.e. you can simply multiply the probabilities.

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    $\begingroup$ In my data it's definitely the case that the difference in quality between the two teams varies considerably between trials. Sometimes the favorite is only a slight favorite (~51% chance), and sometimes it is a huge favorite (~99% chance), and sometimes in between. Since as you mention that would cause a problem for modelling as binomial, is there a workaround? $\endgroup$ – user1205901 - Reinstate Monica Nov 18 '18 at 21:15
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    $\begingroup$ In my opinion the model is simply inappropriate. But I do not have enough experience/knowledge to give a proper answer to whether there is a workaround. $\endgroup$ – Sebastian Nov 18 '18 at 21:24
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I will build on the good insights provided already by Sebastian.

Think of a manufacturing plant that produces widgets. Each widget is either defect-free with probability p or defective with probability (1-p). We can think of this manufacturing plant as some sort of a Bernoulli machine, because the production of each widget is a Bernoulli trial with probability of success p. Pick a sample of n widgets and calculate the fraction of them that are defect-free, let's call it p̂. Because p̂ is obtained by dividing the binomial variable (number of defect-free widgets in a sample of n) by a scalar n, p̂ follows the binomial distribution. By knowing the properties of the sampling distribution of a binomial variable, we can infer from the p̂ we calculated in our sample to the population probability, p. In so doing, we come up with an estimate for the defect-free probability of our Bernoulli machine.

By analogy, you are looking at a manufacturing plant where ex-ante the probability of a widget being defect-free differs from trial to trial ("Sometimes the favorite is only a slight favorite (~51% chance), and sometimes it is a huge favorite (~99% chance)"). We cannot call these Bernoulli trials because the probability p is not the same in each trial. Because a binomial variable is the number of successes in a set of Bernoulli trials, you do not have a binomial variable to speak of because you do not have Bernoulli trials in the first place. Therefore, in your case to model the data as binomial would be to incorrectly assume you have a series of Bernoulli trials, which is not the case.

Even if you picked only one team like Real Madrid, it would appear to be a stretch to think of each of Real Madrid's games as a Bernoulli trial, because they are more likely to win against a weak team than they are against another strong team like Barcelona. Even if you Picked only the Real Madrid games against Barcelona, across time, these would not be Bernoulli trials either because the strength of the teams vary over time, and hence the probability of Real Madrid winning against Barcelona varies over time as well.

Bernoulli trials appear to be rare outside the statistician's lab, and outside the casino (coin flips, throws of dice). Since Bernoulli trials are the atoms that make up a Binomial variable, when modelling a variable as Binomial one should start by making sure that one has valid Bernoulli trials in the first place.

In fact, even the manufacturing plant analogy above is a stretch in terms of it being a Bernoulli machine because in a manufacturing plant we can imagine something like a fuse burning, which would cause the Bernoulli factory to produce defective widgets, one after another, until the fuse is fixed. In the above scenario, it would be a stretch to think of the manufacturing plant as a Bernoulli machine because the probability that the next widget is defect-free is lower if the last widget was defective (because the same burnt fuse may affect the next one), thus the trials are not independent, thus they are not Bernoulli trials. Only if we caveat the description of the defects introduced as those caused by one off events (e.g. an insect landing on a widget during the production process resulted in it being defective, but the next widget is back to having the same ex-ante probability of being defect-free) can we think of the manufacturing plant as a Bernoulli machine, and of the number of successes in n of these trials as a Binomial variable.

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