Visualizing Stirling’s Approximation With Highcharts

July 5, 2016
By

(This article was first published on Ripples, and kindly contributed to R-bloggers)

I said, “Wait a minute, Chester, you know I’m a peaceful man”, He said, “That’s okay, boy, won’t you feed him when you can” (The Weight, The Band)

It is quite easy to calculate the probability of obtaining the same number of heads and tails when tossing a coin N times, and N is even. There are 2^{N} possible outcomes and only C_{N/2}^{N} are favorable so the exact probability is the quotient of these numbers (# of favorable divided by # of possible).

There is another way to approximate this number incredibly well: to use the Stirling’s formula, which is 1/\sqrt{\pi\cdot N/2}

This plot represents both calculations for N from 2 to 200:

Stirling

Although for small values of N, Stirling’s approximation tends to overestimate probability …

Stirling 2

… is extremely precise as N becomes bigger:

Stirling 3

James Stirling published this amazing formula in 1730. It simplifies the calculus to the extreme and also gives a quick way to obtain the answer to a very interesting question: How many tosses are needed to be sure that the probability of obtaining the same number of heads and tails is under any given threshold? Just solve the formula for N and you will obtain the answer. And, also, the formula is another example of the presence of pi in the most unexpected places, as happens here.

Just another thing: the more I use highcharter package the more I like it.

This is the code:

library(highcharter)
library(dplyr)
data.frame(N=seq(from=2, by=2, length.out = 100)) %>%
  mutate(Exact=choose(N,N/2)/2**N, Stirling=1/sqrt(pi*N/2))->data
hc <- highchart() %>% 
  hc_title(text = "Stirling's Approximation") %>% 
  hc_subtitle(text = "How likely is getting 50% heads and 50% tails tossing a coin N times?") %>% 
  hc_xAxis(title = list(text = "N: Number of tosses"), categories = data$N) %>% 
  hc_yAxis(title = list(text = "Probability"), labels = list(format = "{value}%", useHTML = TRUE)) %>% 
  hc_add_series(name = "Stirling", data = data$Stirling*100,  marker = list(enabled = FALSE), color="blue") %>% 
  hc_add_series(name = "Exact", data = data$Exact*100,  marker = list(enabled = FALSE), color="lightblue") %>% 
  hc_tooltip(formatter = JS("function(){return ('Number of tosses: '+this.x+'
Probability: '+Highcharts.numberFormat(this.y, 2)+'%')}")) %>% hc_exporting(enabled = TRUE) %>% hc_chart(zoomType = "xy") hc

To leave a comment for the author, please follow the link and comment on their blog: Ripples.

R-bloggers.com offers daily e-mail updates about R news and tutorials on topics such as: Data science, Big Data, R jobs, visualization (ggplot2, Boxplots, maps, animation), programming (RStudio, Sweave, LaTeX, SQL, Eclipse, git, hadoop, Web Scraping) statistics (regression, PCA, time series, trading) and more...



If you got this far, why not subscribe for updates from the site? Choose your flavor: e-mail, twitter, RSS, or facebook...

Comments are closed.

Search R-bloggers


Sponsors

Never miss an update!
Subscribe to R-bloggers to receive
e-mails with the latest R posts.
(You will not see this message again.)

Click here to close (This popup will not appear again)