Binomial Distribution | Introduction and examples

What Is Binomial

Video tutorial: Wrath of Math, 2017, YouTube

Binomial is a simplistic polynomial.
It has two terms, which could be numeral, variable, or combined element.
for example:
$3x +4$
$x+5$
$x^2 + 5x$

As you can see, they are all binomials.

What Is Binomial Experiment

A binomial experiment is a statistical experiment that has the following properties:

• The experiment consists of n repeated trials.
• Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure.
• The probability of success, denoted by P, is the same on every trial.
• The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials.
  Cite: Stat Trek^[1]

For example, the experiment about flipping a coin and counting the probabilities of each side.

Binomial Distribution

A binomial random variable is the number of successes x in n repeated trials of a binomial experiment. The probability distribution of a binomial random variable is called a binomial distribution. ^[1:1]

For example, when we flip a coin two times,
We can have the results:

Result	Probability
Head * 2	25%
Head, Tail	50%
Tail * 2	25%

$$
b(x; n, P) = C_{n}^{x} * P^ x (1 - P)^{(n - x)}
$$
$$
C_{n}^{x} = \frac{ n!}{ x! (n - x)! }
$$

The properties of the Binomial Distribution:

• The mean of the distribution $μx$ is equal to $n \times P$ .
• The variance $σ2x$ is $nP( 1 - P) $.
• The standard deviation $σx$ is $\sqrt{nP( 1 - P ) }$.
  AND
• $x$: The number of successes that result from the binomial experiment.
• $n$: The number of trials in the binomial experiment.
• $P$: The probability of success on an individual trial.
• $Q$: The probability of failure on an individual trial. (This is equal to 1 - P.)
• $n!$: The factorial of n (also known as n factorial).
• $b(x; n, P)$: Binomial probability - the probability that an $n$ trial binomial experiment results in exactly $x$ successes, when the probability of success on an individual trial is $P$.
• $C_n^x$: The number of combinations of $n$ things, taken $x$ at a time.
  Cite: Stat Trek^[1:2]

Example 1:

Suppose a die is tossed 5 times. What is the probability of getting exactly 2 fours?^[1:3]

So, we have 5 times, which means $n = 5$;
We need exactly four * 2, which means $x = 2$;
The probability of a single trial is 1/6, which give $P = 1/6$

According to the function above, we can have:
$b(2; 5, 1/6) = C_{5}^{2} * (1/6)^ 2 (1 - (1/6))^{(5 - 2)}$

In R, we can calculate the binomial with the function dbinom as dbinom(2, 5, 1/6).
Now we have the results:
$b(2; 5, 1/6) = 0.160751$

Example 2:

$b(x; n, P) = \frac{ n!}{ x! (n - x)!} \times P^ x (1 - P)^{(n - x)}$

Let’s back to the coin.
When we flip it once, and the chances of Head is obviously 50%.
Which is:

$b(1; 1, 1/2) = \frac{ 1!}{ 1! (1 - 1)!} * 0.5^ 1 (1 - 1)^{(1 - 1)}$
$b(1; 1, 1/2) = 1 * 0.5 * 1$
$b(1; 1, 1/2) = 0.5$

When we flip it twice, we can have 4 results: HH, HT, TH, TT

• the chance we have only one head is dbinom(1, 2, 1/2), which is 50%. (HT, TH)
• the chance we have two heads is dbinom(2, 2, 1/2), which is 25%. (HH)
• the chance we have heads from 0 to all is:
  0.25, 0.5, 0.25, which means
  $1: 2:1$
  Now, let’s try to flip it triple times, so we have 8 results:

HHH, HHT, HTH, THH
HTT, THT, TTH, TTT

• one head: HTT, THT, TTH: $3/8 = 0.375$
  $b(1; 3, 1/2) = \frac{ 3!}{ 1! (3 - 1)!} \times 0.5^ 1 (1 - 0.5)^{(3 - 1)}$
  $b(1; 3, 1/2) = 3 \times 0.5 \times 0.25$
  $b(1; 3, 1/2) = 0.375$

• two head: HHT, HTH, THH: $3/8 = 0.275$
  $b(2; 3, 1/2) = \frac{ 3!}{ 2! (3 - 2)!} \times 0.5^ 2 (1 - 0.5)^{(3 - 2)}$
  $b(2; 3, 1/2) = 3 \times 0.25 \times 0.5$
  $b(2; 3, 1/2) = 0.375$

• the chance we have head from 0 to all is:
  0.125, 0.375, 0.375, 0.125, which means
  $1:3:3:1$

With the increase of the flipping number to 10, for instance, we can have the ratio:

for( n in c(1:10)){
  Result = c()
  for(x in c(0:n)){
    Result = c(Result, dbinom(x, n, 1/2)/ dbinom(0, n, 1/2))
  }
  print(Result)
}

                 1
                1 1
               1 2 1
             1  3 3  1
            1 4  6  4 1
          1 5  10  10  5 1
        1 6  15  20  15  6 1
      1 7  21  35  35  21  7 1
    1 8  28  56  70  56  28  8 1
  1 9  36  84 126 126  84  36 9 1
1 10 45 120 210 252 210 120 45 10 1

Poisson theorem^[2]

When The $n$ is infinite large, we can have:
$$
\lim_ {n \to \infty} C_ n ^x P^ x (1 - P)^{(n - x)}, x = 0, 1, 2, …
$$
Let’s say, $nP = \lambda $
When the $n$ is very large and the $P$ is infinite small, we can have:
$$
C_ n ^x P^ x (1 - P)^{(n - x)} \approx \frac{\lambda ^ x}{x!} e^ {-\lambda}
$$

I have no idea how it works. So, I’m gonna stopping here.

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1. Stat_Trek, Binomial Probability Distribution ↩︎ ↩︎ ↩︎ ↩︎

2. 蔡高玉, 2016, 浅谈二项分布及其应用, 《考试周刊》 2016年A5期 期刊 ↩︎