population.data.set <- c( rep( 0, 20 ), rep( 1, 18 ) )

print( population.data.set )

# [1] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

n.repetitions <- 5000

set.seed( 2717282 )

sample.data.set <- sample( population.data.set, n.repetitions, replace = T )

cumulative.proportion <- cumsum( sample.data.set ) / ( 1:n.repetitions )

print( cbind( 1:10, sample.data.set[ 1:10 ], 
  cumulative.proportion[ 1:10 ] ) )


#       [,1] [,2]      [,3]
#  [1,]    1    0 0.0000000
#  [2,]    2    0 0.0000000
#  [3,]    3    0 0.0000000
#  [4,]    4    0 0.0000000
#  [5,]    5    0 0.0000000
#  [6,]    6    1 0.1666667
#  [7,]    7    1 0.2857143
#  [8,]    8    1 0.3750000
#  [9,]    9    0 0.3333333
# [10,]   10    0 0.3000000

plot( 1:n.repetitions, cumulative.proportion, type = 'l',
  lwd = 1, col = 'blue', xlab = 'repetition number', lty = 2,
  ylab = 'cumulative proportion of red roulette numbers', ylim = c( 0, 1 ) )

abline( h = 18 / 38, lwd = 3, col = 'black' )

n.simulations <- 100

for ( i in 2:n.simulations ) {

  sample.data.set <- sample( population.data.set, n.repetitions, replace = T )

  cumulative.proportion <- cumsum( sample.data.set ) / ( 1:n.repetitions )

  lines( 1:n.repetitions, cumulative.proportion, lwd = 1, lty = 2,
    col = 'blue' )

  Sys.sleep( 0.5 )

}

abline( h = 18 / 38, lwd = 3, col = 'black' )