Ten days of statistics (5) - Binomial distribution

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Binomial experiment

A binomial experiment is an experiment that is:

The experiment consists of $n$ repeated trials
The trials are independent
The outcome of each trial is either success or failure

The sample space of a binomial experiment only has 2 points, 1 (success) and 0 (failure). Let $p(x)$ be the probability of an experiment ending with result $x$

p(x) = \begin{cases} 1-k & \text{ if x = 0 } \\ k & \text{ if x = 1 } \\ 0 & \text{ otherwise } \end{cases}

Binomial distribution

We define a binomial process to be a binomial experiment meeting the following conditions:

The number of successes is $x$ .
The total number of trials is $n$ .
The probability of success of $1$ trial is $p$ .

Let $b(x,n,p)$ is the probability of having exact $x$ successes out of $n$ trials. Then we have

b(x, n, p) = C \binom{n}{x} \times p^x \times (1-p)^{n-x}

Note: $C\binom{n}{r} = n! \div (r! \times (n-r)!)$

Example

Toss a coin 10 times. Let’s find the following probabilities:

Getting 5 heads
Getting at least 5 heads
Getting at most 5 heads

Using binomial distribution formular, we have

b(5, 10, 0.5) = 0.24609375

We have $\approx 25\%$ chance of getting exact 5 heads.

\sum_{x=5}^{10} b(x,10,0.5) = 0.623046875

We have $\approx 62\%$ chance of getting at least 5 heads

\sum_{x=0}^5 b(x, 10, 0.5) = 0.623046875

We also have $\approx 62\%$ chance of getting at most 5 heads

Negative binomial experiment

A negative binomial experiment is an experiment that is:

The probability of success of $1$ trial is $p$
The trials continue until $x$ successes are observed
The total number of trials is not fixed

Note: the different between negative binomial experiment and binomial experiment is on the interest. Negative version only interest in getting exact $x$ successes, normal version interest in performing exact $n$ trials.

Negative binomial distribution

Formally speaking, let $b^*(x, n, p)$ be the probability of having

Exact $x-1$ successes after $n-1$ trials
Exact $x$ successes after $n$ trials

b^*(x, n, p) = C \binom{n-1}{x-1} \times p^x \times (1-p)^{n-x}

Geometric distribution

The geometric distribution is a special case of the negative binomial distribution that deals with the number of trials required to get a success (i.e. counting the number of failures before the first success).

g(n,p) = p \times (1-p)^{n-1}

Example

Vova’s family lives in Russia where the weather is cold. The chance for Vova’s wife to give birth to a boy is 30%. What is the probability that Vova have his first son at his fifth child?

For this experiment, $n = 5, p = 0.3$ , we have $g(5, 0.3) = 0.3 \times 0.7^4 = 0.07203$ . So Vova has $\approx 7\%$ of having his first son at his fifth child.

Practice

Hackerrank has some exercises for you to test your knowledge:

Next lesson: Poisson & normal distribution