1.

plot(x    = 0:20, 
     y    = dbinom(x    = 0:20, 
                   size = 20, 
                   prob = .5
                  ), 
     type = 'h',
     main = 'Binomial Distribution (n=20, p=0.5)',
     ylab = 'Probability',
     xlab = '# Successes',
     lwd  = 3
     )

dbinom(x = 9:12, size = 20, prob = .5)
## [1] 0.1601791 0.1761971 0.1601791 0.1201344
sum(dbinom(x = 9:12, size = 20, prob = .5))
## [1] 0.6166897
round(x = sum(dbinom(x = 9:12, size = 20, prob = .5)),digits = 4)
## [1] 0.6167

The probability is 0.6167.

plot(x    = 0:13, 
     y    = dbinom(x    = 0:13, 
                   size = 13, 
                   prob = .2
                  ), 
     type = 'h',
     main = 'Binomial Distribution (n=13, p=0.2)',
     ylab = 'Probability',
     xlab = '# Successes',
     lwd  = 3
     )

dbinom(x = 4:5, size = 13, prob = .2)
## [1] 0.15354508 0.06909529
sum(dbinom(x = 4:5, size = 13, prob = .2))
## [1] 0.2226404
round(x = sum(dbinom(x = 4:5, size = 13, prob = .2)),digits = 4)
## [1] 0.2226

The probability is 0.2226 .

3.

round(ppois(3,4.2),4)
## [1] 0.3954

The probability is 0.3954 .

4.

N <- 17
K <- 6
n <- 3
k_values <- 0:1

# Calculate hypergeometric probabilities for each k
probabilities <- dhyper(k_values, K, N-K, n)

# Sum the probabilities
total_probability <- sum(probabilities)
ans <- round(total_probability, 4)
print(ans)
## [1] 0.7279

The probability is 0.7279.

1-round(phyper(q = 1, m = 6, n = 19, k = 6),4)
## [1] 0.4529

The probability is 0.4529.

round(pnorm(1460, 800, 300) - pnorm(1040, 800, 300),4)
## [1] 0.198

The probability is 0.1980.

round(pnorm(111,106, 4) - pnorm(103, 106, 4),4)
## [1] 0.6677

The probability is 0.6677 . 8.

round(qnorm(0.03, mean = 3.34, sd = 0.07),2)
## [1] 3.21
round(qnorm(0.03, mean = 3.34, sd = 0.07,lower.tail = F),2)
## [1] 3.47

Top 3% is 3.47 and bottom 3% is 3.21.

round(qnorm(0.09,75.8,8.1,lower.tail = F),0)
## [1] 87

The minimum is 87.

mu = 155*0.61
sd = (155*0.61*(1-0.61))^0.5
round(dnorm(96,mu,sd),4)
## [1] 0.0639

The probability is 0.0639.