Typically the binomial distribution is introduced by using coin tosses. The binomial is used to calculate the number of successes (head or tails whichever you prefer) in a designated number of flips. The problem with this is that the probability is symmetric. By that I mean the probability for a success is the same as that for a failure and this can result in some confusion and so I am going to use asymmetric examples where the probabilities are different.
The binomial distribution gives the probability of the number of successes (k) from n Bernoulli trials. Where a Bernoulli trial is a random experiment with two possible outcomes that are defined as success or failure.
These trials are named after Jacob Bernoulli (there are a lot of Bernoulli family members involved in maths).
It is defined by two parameters, n - the number of trials and p the probability of a success.
\[ f(k,n,p)= P(X=k) = {n \choose k }p^{k}(1-p)^{n-k} \]
Sometimes 1-p is written as q.
The binomial distribution is:
A discrete random variable (total probability is 1)
The probability of a success is constant.
Only valid for a fixed number of trials (n)
We have already seen an example of the binomial distribution in practice in the example I gave for a discrete random variable, where a success was finding a cytosine in a window of 3 nucleotides of sequence if all the nucleotides have the same abundance (n=3 for that example and p=0.25). This was also used to prove that the distribution is a random variable and that the probabilities all add up to one.
Rather than using the formula it is easier to use the built in binomial distribution function within R to calculate the probabilities.
From the previous calculations:
P(X=0) = P(3 [A,T,G]) = \((\frac{3}{4})^{3} = \frac{27}{64}\)
P(X=1) = P(C, 2 [A,T,G]) = \(3(\frac{1}{4}(\frac{3}{4})^{2}) = \frac{27}{64}\)
P(X=2) = P(CC, [ATG]) = \(3((\frac{1}{4})^{2}\frac{3}{4})=\frac{9}{64}\)
P(X=3) =P(CCC) = \((\frac{1}{4})^{3} = \frac{1}{64}\)
library(ggplot2)
n <- c(0:3)
p <- c(27/64,27/64,9/64,1/64)
data <- data.frame(n,p)
plot <- ggplot(data=data, aes(x=n, y=p))+
geom_bar(stat="identity", fill="cornflowerblue") +
labs(title="Probability Distribution for the Number of \n Cytosines Found in a Three Nucleotide Window Assuming Equal Abundance", x="Number of Cytosines", y="Probability")
plot
library(ggplot2)
x <- c(0:3)
y <- dbinom(x, 3, 0.25)
data <- data.frame(x,y)
plot <- ggplot(data=data, aes(x=x, y=y))+
geom_bar(stat="identity", fill="cadetblue") +
labs(title="Probability Distribution for the Number of \n Cytosines Found in a Three Nucleotide Window Assuming Equal Abundance", x="Number of Cytosines", y="Probability")
plot
This is exactly the same plot as we calculated before.
It is known that 20% of a population of laboratory mice are infertile.
a) if 5 mice are chosen at random, find the probability that exactly 2 of them are infertile
b) Find the least number of mice that need to chosen for the probability of choosing at least one infertile mouse to be greater than 0.99
For part a we can use R
dbinom(2, 5, 0.2)[1] 0.2048For the second part we have to calculate the number of trials which is a bit more complex.
\(P(X \ge 1) > 0.99 = 1 - P(X=0) = 1-(0.8)^{n}\)
\((0.8)^{n} < 0.01\)
\(n log(0.8) < log(0.01)\)
\(n > \frac{log(0.01)}{log(0.8)}\)
log(0.01)/log(0.8)[1] 20.6377So \(n \ge 21\)
dbinom(4,4,0.5)[1] 0.0625To calculate this you need to work out the probability of having 2 or less blemished apples and subtract it from 1.
1-pbinom(2,10,0.15)[1] 0.1798035dbinom(0,5,0.1)[1] 0.59049P(X>1) > 0.9
\((0.9)^{n}<0.1\)
\(n log(0.9) < log(0.1)\)
log(0.1)/log(0.9)[1] 21.85435\(n \ge 22\)
These calculations are VERY IMPORTANT for sample size calculations when you have a population containing a small number of the people affected by the condition that you want to study. If the numbers are very low then a case-control study makes more sense rather than a cohort study.
The final part of this series is going to cover the geometric/exponential distribution and the Poisson Distribution which are closely connected to one another.
library(ggplot2)
library(RColorBrewer)
library(knitr)
x <- c(0:4)
success <- x
success <- as.factor(success)
prob <- dbinom(x, 4, 0.2)
data <- data.frame(success,prob)
#| label: tbl-binomial1
#| tbl-cap: "Binomial n=4 p=0.2 Distribition"
#| tbl-colwidths: [10,20]
#| echo: fenced
kable(data, col.names = c("Successes","Probability") )| Successes | Probability |
|---|---|
| 0 | 0.4096 |
| 1 | 0.4096 |
| 2 | 0.1536 |
| 3 | 0.0256 |
| 4 | 0.0016 |
p <- ggplot(data=data, aes(x=success, y=prob, fill=success))+
geom_bar(stat="identity", color="black") +
scale_fill_brewer(palette = "Blues") +
theme(legend.position="none") +
labs(title="The n=4 p=0.2 Binomial Distribition" , x="Successes", y="Probability")
p
library(ggplot2)
library(RColorBrewer)
library(knitr)
x <- c(0:4)
success <- x
success <- as.factor(success)
prob <- dbinom(x, 4, 0.5)
data <- data.frame(success,prob)
#| label: tbl-binomial1
#| tbl-cap: "Binomial n=4 p=0.5 Distribition"
#| tbl-colwidths: [10,20]
#| echo: fenced
kable(data, col.names = c("Successes","Probability") )| Successes | Probability |
|---|---|
| 0 | 0.0625 |
| 1 | 0.2500 |
| 2 | 0.3750 |
| 3 | 0.2500 |
| 4 | 0.0625 |
p <- ggplot(data=data, aes(x=success, y=prob, fill=success))+
geom_bar(stat="identity", color="black") +
scale_fill_brewer(palette = "Purples") +
theme(legend.position="none") +
labs(title="The n=4 p=0.5 Binomial Distribition" , x="Successes", y="Probability")
p
library(ggplot2)
library(RColorBrewer)
library(knitr)
x <- c(0:3)
success <- x
success <- as.factor(success)
prob <- dbinom(x, 3, 0.4)
data <- data.frame(success,prob)
#| label: tbl-binomial1
#| tbl-cap: "Binomial n=3 p=0.4 Distribition"
#| tbl-colwidths: [10,20]
#| echo: fenced
kable(data, col.names = c("Successes","Probability") )| Successes | Probability |
|---|---|
| 0 | 0.216 |
| 1 | 0.432 |
| 2 | 0.288 |
| 3 | 0.064 |
p <- ggplot(data=data, aes(x=success, y=prob, fill=success))+
geom_bar(stat="identity", color="black") +
scale_fill_brewer(palette = "Greens") +
theme(legend.position="none") +
labs(title="The n=3 p=0.4 Binomial Distribition" , x="Successes", y="Probability")
p
library(ggplot2)
library(RColorBrewer)
library(knitr)
x <- c(0:7)
success <- x
success <- as.factor(success)
prob <- dbinom(x, 7, 0.4)
data <- data.frame(success,prob)
#| label: tbl-binomial1
#| tbl-cap: "Binomial n=7 p=0.4 Distribition"
#| tbl-colwidths: [10,20]
#| echo: fenced
kable(data, col.names = c("Successes","Probability") )| Successes | Probability |
|---|---|
| 0 | 0.0279936 |
| 1 | 0.1306368 |
| 2 | 0.2612736 |
| 3 | 0.2903040 |
| 4 | 0.1935360 |
| 5 | 0.0774144 |
| 6 | 0.0172032 |
| 7 | 0.0016384 |
p <- ggplot(data=data, aes(x=success, y=prob, fill=success))+
geom_bar(stat="identity", color="black") +
scale_fill_brewer(palette = "Oranges") +
theme(legend.position="none") +
labs(title="The n=7 p=0.4 Binomial Distribition" , x="Successes", y="Probability")
p
library(ggplot2)
library(RColorBrewer)
library(knitr)
x <- c(0:10)
success <- x
success <- as.factor(success)
prob <- dbinom(x, 10, 0.4)
data <- data.frame(success,prob)
#| label: tbl-binomial1
#| tbl-cap: "Binomial n=10 p=0.4 Distribition"
#| tbl-colwidths: [10,20]
#| echo: fenced
kable(data, col.names = c("Successes","Probability") )| Successes | Probability |
|---|---|
| 0 | 0.0060466 |
| 1 | 0.0403108 |
| 2 | 0.1209324 |
| 3 | 0.2149908 |
| 4 | 0.2508227 |
| 5 | 0.2006581 |
| 6 | 0.1114767 |
| 7 | 0.0424673 |
| 8 | 0.0106168 |
| 9 | 0.0015729 |
| 10 | 0.0001049 |
p <- ggplot(data=data, aes(x=success, y=prob, fill=success))+
geom_bar(stat="identity", color="black") +
scale_fill_brewer(palette = "BrBG") +
theme(legend.position="none") +
labs(title="The n=10 p=0.4 Binomial Distribition" , x="Successes", y="Probability")
p