Probability Distributions

Binomial Probability Function

Suppose we perform

n

Bernoulli trials, where the probability of success in each trial is

p\in (0,1)

. We can define the random variable

X

as the number of successes obtained in the

n

trials. The sample space of this experiment will consist of all sequences of length

n

of successes and failures, and therefore its cardinality will be

2^n

. The values that

X

can take are

0,1,\cdots,n

, and it is said that

X

follows a binomial distribution with parameters

n

and

p

. This is written as

X \sim \text{bin}(n,p)

, and its probability function is:

f(x) = \left\{ \begin{align*} \binom{n}{x} p^x (1-p)^{n-x} & \hspace{1cm} \text{ if }~~ x = 0, 1, \dots, n \\ 0 & \hspace{1cm} \text{otherwise} \end{align*} \right.

where

\binom{n}{x} = \frac{n!}{x!(n-x)!}

is the binomial coefficient.

This animation consists of a pyramid with $n$ rows, each containing obstacles. These obstacles cause the ball, when it hits them, to move to the left (failure) with a probability of $1-p$ or to the right (success) with a probability of $p$ .

In this way, the values of $X=0,1,\ldots,n$ represent the number of successes obtained, that is, the number of times the ball moved to the right.

(You might need to reload the page for each animation)

p =

0.5

n =