#generate a single random variable that represents the number of 
#heads in 100 flips of our unfair coin
rbinom(1, size = 100, prob = 0.7)

## [1] 66

#request 100 observations, each of size 1, with success probability of 0.7
flips2 <- rbinom(100, size = 1, prob = 0.7)
flips2

##   [1] 1 1 1 1 1 1 0 1 1 1 0 0 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 1 0 0 0 0 1
##  [36] 1 1 1 0 0 1 1 1 1 1 0 1 0 0 0 1 1 1 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 0
##  [71] 0 0 1 1 1 1 0 1 0 0 0 0 1 0 1 1 0 1 1 1 0 1 1 1 1 0 0 0 0 1

## Compute P(45 < X < 55) for X Binomial(100,0.5)
sum(dbinom(46:54, size = 100, prob = 0.5))

## [1] 0.6317984

n <- 2000
k <- seq(0, n, by = 20)
plot (k, 
      dbinom(k, size = n, prob = pi/10, log = TRUE), 
      type = "l", 
      ylab = "log density",
      main = "dbinom(*, log=TRUE) is better than  log(dbinom(*))")

lines(k, log(dbinom(k, n, pi/10)), col = "red", lwd = 2)