Assignment 6th Oct

N <- 100
M <- 6

# Part (a)
for(K in 1:100) {
  prob_accept <- dhyper(0, M, N-M, K)
  if(prob_accept < 0.10) {
    cat("K =", K, "Prob =", round(prob_accept, 6), "\n")
    break
  }
}

## K = 32 Prob = 0.091819

# Part (b)  
for(K in 1:100) {
  prob_accept <- dhyper(0, M, N-M, K) + dhyper(1, M, N-M, K)
  if(prob_accept < 0.10) {
    cat("K =", K, "Prob =", round(prob_accept, 6), "\n")
    break
  }
}

## K = 51 Prob = 0.093314

n_customers <- 1000
p <- 0.5
confidence <- 0.99

# Part (a)
N_binomial <- qbinom(confidence, n_customers, p)
 N_binomial

## [1] 537

# Part (b)
mu <- n_customers * p
sigma <- sqrt(n_customers * p * (1 - p))
z_value <- qnorm(confidence)
N_normal <- ceiling(mu + z_value * sigma)
N_normal

## [1] 537

#part a
 n <- 60
 p <- 1/90
 1 - pbinom(4, n, p)

## [1] 0.000556628

 # Part b
 n_schools_ny <- 310
 p_school <- 0.0006
 p_at_least_one_ny <- 1 - dbinom(0, n_schools_ny, p_school)
 cat("Probability at least one school in NY has the event:", p_at_least_one_ny, "\n")

## Probability at least one school in NY has the event: 0.1697728

#part c
 n_states <- 500
 p_state <- p_at_least_one_ny  
 p_at_least_one_national <- 1 - dbinom(0, n_states, p_state)
 cat("Probability at least one state has the event in 10 years:", p_at_least_one_national, "\n")

## Probability at least one state has the event in 10 years: 1