# STAT 515 Lec 05

# Computing Binomial probabilities

n <- 10
x <- 3
p <- 0.7
  
# choose(10,3) * (0.7)^3 *(1 - 0.7)^(10 - 3)
choose(n,x)*p^x*(1-p)^(n-x)

# P(X <= 6)

x <- c(0,1,2,3,4,5,6)
choose(n,x)*p^x*(1-p)^(n-x)

sum( choose(n,x)*p^x*(1-p)^(n-x) )



### hypergeometric distribution

N <- 100
M <- 10
n <- 5
x <- 3

choose(M,x) * choose(N-M,n-x) / choose(N,n)

## The spades example;

N <- 52
M <- 13
n <- 5
x <- 0:5 # all whole numbers 0 through 5

px <- choose(M,x)*choose(N - M, n -x) / choose(N,n)
px

# get binomial probabilities (for sampling with replacement)

p <- 13/52
n <- 5
x <- 0:5
px_binom <- choose(n,x) * p^x *(1 - p)^(n-x)
px_binom


round(px,3)
round(px_binom,3)

# mix ten deck of cards together
N <- 520
M <- 130
n <- 5
x <- 0:5 # all whole numbers 0 through 5

px2 <- choose(M,x)*choose(N - M, n -x) / choose(N,n)
px2


round(px,3) # single deck
round(px2,3) # combined deck
round(px_binom,3) # binomial (with replacement)


# suppose I want P(X <= 3)

x <- 0:3
sum(choose(M,x)*choose(N - M, n -x) / choose(N,n))

# for binomial

x <- 0:3
sum( choose(n,x) * p^x *(1 - p)^(n-x))
pbinom(3,n,p) # easier

#P (X = 3) , X is binomial

dbinom(3,n,p)

# HW 3, Q3, part d

N <- 100
d <- 0:10
n <- 10
px0 <- choose(d,0)*choose(N - d,n) / choose(N,n)
px0