discussion_9

pg. 338
2. Let S200 be the number of heads that turn up in 200 tosses of a fair coin. 
Estimate
  (a) P(S200 =100). 
  (b) P(S200 =90). 
  (c) P(S200 =80).

# Number of tosses
n <- 200

# Probability of getting a head
p <- 0.5

# Calculate the mean and standard deviation of the binomial distribution
mu <- n * p
sigma <- sqrt(n * p * (1 - p))

exp_P <- function(x) {
  z <- (x - mu) / sigma
  (P <- dnorm(x, mean = mu, sd=sigma)) 
#The dnorm function takes in three arguments: x, mean, and sd, which represent the value at which to evaluate the density, the mean of the distribution, and the standard deviation of the distribution, respectively.
}

  (a) P(S200 = 100)

exp_P(100)

## [1] 0.05641896

# The estimated probability of getting exactly 100 heads in 200 tosses of a fair coin

  (b) P(S200 = 90)

exp_P(90)

## [1] 0.02075537

# The estimated probability of getting exactly 90 heads in 200 tosses of a fair coin

  (c) P(S200 = 80)

exp_P(80)

## [1] 0.001033349

# The estimated probability of getting exactly 80 heads in 200 tosses of a fair coin