This histogram is the result of running a thousand repetitions of 1127422 trials and record the longest number of losses in a row. From the simulated results, we can say that the probability of less than 16 in a row and more than 31 in a row is less than one in a thousand. So I'd say 65 in a row is significantly improbable.
Since I'm home now I've looked for the paper I thought I had but can't find it. The histogram looks familiar though and I think it might be an extreme value distribution, but I'm not sure. I'll keep looking.
If you want to try the simulation for yourself, especially if you have a faster computer than my macbook air and want to try more than a thousand repetitions, here's the R script I wrote:
Code:
require(ggplot2)
# function to generate simulated maximum losses in a row
sim.func <- function(x) {
#create x trials: win = 1, loss = 0
# and find all wins and losses in a row
a <- rle(rbinom(x, 1, 32000/65536))
# find the longest number of losses in a row
b <- max(a$lengths[which(a$values == 0)])
# if no losses, return 0 otherwise return b
if(b == -Inf){return(0)} else {return(b)}
}
# number of trials
trials <- 1127422
# number of repetition of trials
# change for greater accuracy
repetitions <- 1000
# generate results
result <- sapply(rep(trials, repetitions), sim.func)
result.df <- data.frame(count = result)
# generate plot
plot.1 <- ggplot(result.df, aes(x=count)) +
geom_bar(binwidth=1) +
scale_x_continuous(breaks=seq(min(result.df),max(result.df),2)) +
labs( x = "", y = "Count",
title = "Histogram of simulated maximum number of sequential losses:\n1000 repetitions of 1127422 trials at P(win) = 32000/65536") +
xlab("") + ylab("count")
# display plot
plot.1
#eof
# function to generate simulated maximum losses in a row
sim.func <- function(x) {
#create x trials: win = 1, loss = 0
# and find all wins and losses in a row
a <- rle(rbinom(x, 1, 32000/65536))
# find the longest number of losses in a row
b <- max(a$lengths[which(a$values == 0)])
# if no losses, return 0 otherwise return b
if(b == -Inf){return(0)} else {return(b)}
}
# number of trials
trials <- 1127422
# number of repetition of trials
# change for greater accuracy
repetitions <- 1000
# generate results
result <- sapply(rep(trials, repetitions), sim.func)
result.df <- data.frame(count = result)
# generate plot
plot.1 <- ggplot(result.df, aes(x=count)) +
geom_bar(binwidth=1) +
scale_x_continuous(breaks=seq(min(result.df),max(result.df),2)) +
labs( x = "", y = "Count",
title = "Histogram of simulated maximum number of sequential losses:\n1000 repetitions of 1127422 trials at P(win) = 32000/65536") +
xlab("") + ylab("count")
# display plot
plot.1
#eof
