When to use Binomial versus Beta distribution?
A footballer is known to score 70% of the penalty kicks he shoots. In the next season we expect him to shoot 10 penalty kicks, how many of them will he score?
He will score 7 out of the 10 penalty kicks, obviously!
Actually, 10 penalty kicks is a very little number to make a definite conclusion from. This obvious 7 could turn up to be 8 or 9 with a bit of luck, or he can miss a couple of unexpected penalties and the 7 turns out to be 5. Obviously, huh?
With such a small number, there are hardly any obvious answers, we rather need to express our belief in the form of a distribution.
And in this case, it is a binomial distribution that we need.
This is what the above distribution represents:
Say we manage to convince this football player to shoot 10 penalty kicks, then ask him to shoot another 10, then another, up to 1,000 sets of 10 penalty kicks. Each time we calculate how many shots out of the 10 he scored, and create a histogram out of it. That’s what we have here.
Keep in mind, the player can never score 7.7 penalty kicks, or 3.8 kicks, only integers are allowed on the x axis. That’s why the binomial distribution is discrete probability distribution, and this histogram is called probability mass function.
Rather than bothering you with the confusing statistical terms, like experiment, event and success, etc. Let’s use the vocabulary of our example here to explain how the binomial distribution can be used.
When you know the probability (p) of a player scoring a penalty kick. Then you use the binomial distribution to express your belief in how likely they will score (x) penalty kicks out of (k) kicks. As you can see, there are two parameters here, the probability (p) and the number of kicks the player will shoot (k), and from there you can plot the probability mass function in terms of the number of scored kicks (x).
As mentioned, because 10 is a very small number we cannot be confident that the player will score 7 out of those 10 kicks. You can even see in the…