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Binomial Distributions

We now learn the about Binomial Distribution. In particular, in this section, we learn the formula for the binomial density function: \[P \begin{pmatrix} X = r \end{pmatrix} = \begin{pmatrix} n \\ r \end{pmatrix} p^rq^{n-r}\]

When to use Binomial Distributions

We use binomial distribution as soon as we're dealing with several repetitions of a same experiment, we'll say "\(n\) trials", in which there are:

  • two possible outcomes, which should be thought of as success/failure, win/lose, or yes/no.
  • the outcome of each trial of the experiment is independent from the previous/next.

Discrete Random Variable, \(X\)

For such scenarios, we define the Discrete Random Variable \(X\), whose value can be any of the potential number of successes/wins in the experiment.

The binomial density function will allow us to calculate the probability of there being exactly \(r\) successes of the experiment, when repeating it \(n\) times. In other words the binomial density function will allow us to calculate the probability that \(X = r\).

The following example is a typical case of binomial distribution:

A biased coin, for which the probability of flipping tails is \(0.3\). We flip the coin \(5\) times and we wish to find the probability of obtaining exactly (no more no less) \(3\) tails.
To answer such questions we learn the binomial density function formula.

When a discrete random variable follows a binomial distribution, with \(n\) trials, in which the probability of a succees is \(p\), we write: \[X \sim B \begin{pmatrix}n,p \end{pmatrix}\]

Binomial Distribution - Probability Density Function (PDF)

Given an experiment, which satisfies all the criteria listed above and for which the probability of a success is \(p\). The probability of \(r\) successes in \(n\) trials is: \[P \begin{pmatrix} X = r \end{pmatrix} = \begin{pmatrix} n \\r \end{pmatrix}p^r.\begin{pmatrix} 1 - p \end{pmatrix}^{n-r}\] If we define a "loss" as \(q=1-p\), then this formula can also be written: \[P \begin{pmatrix} X = r \end{pmatrix} = \begin{pmatrix} n \\r \end{pmatrix}p^r.q^{n-r}\] Since \(\begin{pmatrix} n \\ r \end{pmatrix} = ^nC_r \) this formula can also be written: \[P \begin{pmatrix} X = r \end{pmatrix} = ^nC_r p^r.q^{n-r}\]

Tutorial

In the following tutorial, we're reminded of this formula and explained how it actually works. Watch it now, by the end of it the binomial distribution formula will be far more meaningful.

Example

We're given a biased coin with which the probability of flipping tails is: \[p = 0.3\] We flip the coing \(5\) times. What is the probability of obtaining exactly \(3\) tails?

Solution/Method

We start by defining each of the following:

  • The discrete random variable, \(X\), as: the number of tails obtained in the experiment.
  • The coin is flipped \(5\) times so the number of trials is \(n=5\).
  • We're interested in the probability of flipping \(3\) tails, so \(r = 3\).
  • The probability of a "success" is equal to the probability of flipping tails, \(p=0.3\).
  • The probability of a "loss" is the complement of \(p\), so that's \(q = 1 - p = 0.7\).
Combining all of this, our binomial distribution formula becomes: \[P \begin{pmatrix} X = 3 \end{pmatrix} = \begin{pmatrix} 5 \\ 3 \end{pmatrix}0.3^3.0.7^{5-3}\] That's: \[P \begin{pmatrix} X = 3 \end{pmatrix} = \begin{pmatrix} 5 \\ 3 \end{pmatrix}0.3^3.0.7^2\] Calculating this leads to: \[\begin{aligned} P \begin{pmatrix} X = 3 \end{pmatrix} & = \begin{pmatrix} 5 \\ 3 \end{pmatrix}0.3^3.0.7^2 \\ & = 10 \times 0.3^3 \times 0.7^2 \\ P \begin{pmatrix} X = 3 \end{pmatrix} & = 0.1323 \end{aligned}\] So there is approcimately a \(13\% \) chance of obtaining exactly \(3\) taims when flipping this coin \(5\) times.

Exercise

  1. Charlotte and Clara are planning a \(7\) day trip to Falkenberg, Sweden. The probability that it rains on any one of those days is \(0.3\).
    What is the probability that it rains during exactly \(3\) of those \(7\) days?

  2. A factory produces bolts that are then sold to car manufacturers. Given there is a \(1\) in \(50\) chance that their machine produces a faulty bolt (that they can't sell) and given that the production of each bolt doesn't affect the production of any other bolt, what is the probability of there being exactly five faulty bolts if they manufacture \(200\) of them?

  3. \(8\) students sit a highly advanced mathematics test. The probability that a student scores above \(80\% \) is \(0.1\).
    What is the probability that exactly \(4\) of the \(8\) students score higher than \(80\% \) ?

  4. James likes to play darts. When he plays, on each throw there is a \(20\%\) chance that he hits a bullseye.
    He throws \(10\) darts in a row. What is the probability that he hits exactly \(8\) bulleyes?

  5. A famous smartphone company has its phones manufactures such that the probability of a phone being faulty is 0.03.
    In an hour \(100\) phones are manufactures. What is the probability that exactly \(4\) of those phones are faulty?

Answers Without Working