The Law of Large Numbers

The law of large numbers is a theorem that describes the result of performing the same experiment a large number of times.According to this law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to be closer as more trials are performed.

$\bar{x_{n}} \to \mu$ as ${n} \to \infty$

Visualizing Law of Large Numbers using coin flips

Bias = 0.5

For a simple example, we will use an unbiased coin.This means that the probability of getting a head/tail is 0.5. As the number of coin tosses increases, the average proportion of heads/tails achieved would be close but not equal to 0.5.

library(ggplot2) library(gganimate) library(tweenr) library(dplyr) n = 500 pHeads =0.5 flipSequence = sample( x=c(0,1),prob = c(1-pHeads,pHeads),size=n,replace=T) r = cumsum(flipSequence) n= 1:n runProp = r/n flip_data <- data.frame(run=1:500,prop=runProp) theme_set(theme_bw()) g <- ggplot(flip_data,aes(x=run,y=prop,frame=run)) +geom_path(aes(cumulative=T))+xlim(1,500)+ylim(0.0,1.0)+geom_hline(yintercept = 0.5)+ggtitle("Running Proportion Heads of Unbiased Coin")+ylab("Proportion of Heads")+xlab("Flip Number") gganimate(g,"attempt_cumulative.mp4",title_frame = F,interval=0.1)

flips_simulation <- function(n,pHeads,seed,name_of_file){ if (!require("ggplot2")) install.packages("ggplot2") if(!require("devtools"))install.packages("devtools") if(!require("gganimate"))install_github("dgrtwo/gganimate") set.seed(seed) flipSequence = sample(x=c(0,1),prob=c(1-pHeads,pHeads),size=n,replace=T) r=cumsum(flipSequence) runProp = r/(1:n) data_flip <- data.frame(index=1:length(runProp),runProp) g <- ggplot(data_flip,aes(x=index,y=runProp,frame=index)) +geom_path(aes(cumulative=T))+xlim(1,n)+ylim(0.0,1.0)+geom_hline(yintercept = pHeads)+ggtitle(paste("Running Proportion Heads of Coin of Bias"),pHeads)+ylab("Proportion of Heads")+xlab("Flip Number") gganimate(g,paste(name_of_file,".mp4",sep=""),title_frame = F,interval=0.1) }

The average proportion of heads eventually converge to 0.5. What about other biases? Before we start, let’s define a function to produce the simulation

In the above function, the inputs are n(Number of Trials),pHeads(probability of head),seed(seed value for reproducibility) and name_of_file(name of your .mp4 file).Now let’s run this function for different values of bias and number of trials.

Bias = 0.6 , n=1000

flips_simulation(1000,0.6,500,"bias6")

Bias = 0.4,n=100

flips_simulation(100,0.4,500,"bias4")