###### R code for calculating probabilities 

### Binomial probabilities

## Example 1:  Y ~ bin(n = 10, p = 0.02)

# What is P(Y >= 2)?

# Note P(Y >= 2) = 1 - P(Y <= 1):

1 - pbinom(1, size = 10, prob = 0.02)

## Example 2:  Y ~ bin(n = 10, p = 0.40)

# What is P(Y <= 7)?

pbinom(7, size = 10, prob = 0.40)

# What is P(Y >= 4)?

# Note P(Y >= 4) = 1 - P(Y <= 3):

1 - pbinom(3, size = 10, prob = 0.40)

## Plots of these two binomial distributions:

# Y ~ bin(n=10, p=0.02):

plot(0:10, dbinom(0:10, size=10, prob=0.02), type='h', lwd=2, xlab='y', ylab='p(y)')

# Y ~ bin(n=10, p=0.40):

windows()

plot(0:10, dbinom(0:10, size=10, prob=0.40), type='h', lwd=2, xlab='y', ylab='p(y)')

# What is the main difference in the distributions?

###############################################################################

### Geometric probabilities


## Example 1: Y ~ geom(p = 0.75)

# What is P(Y = 4)?

dgeom(4-1, prob = 0.75)

# What about P(Y <= 4)?
# This is the probability the person will need
# AT MOST 4 attempts to pass the driving test [not in notes].

pgeom(4-1, prob = 0.75)


### Negative Binomial probabilities


## Example 2: Y ~ NB(r = 3, p = 0.40)

# What is P(Y = 10)?

dnbinom(10-3, size = 3, prob = 0.40)

# What about P(Y <= 10)?
# This is the probability that AT MOST 10 employees must be checked
# to find 3 with asbestos traces [not in notes].

pnbinom(10-3, size = 3, prob = 0.40)

###############################################################################

### Hypergeometric probabilities

## Example 3: Y ~ hyper(r=8, N=20, n=6):

# What is P(Y = 1)?

# Note (N-r) = 12:

dhyper(1, 8, 12, 6)

# What is P(Y <= 1)?
# This is the probability of ONE OR FEWER successes in the sample.

phyper(1, 8, 12, 6)

## Example 1: Y ~ hyper(r=2, N=100, n=10):

# What is P(Y >= 1)?
# This is the probability of ONE OR MORE successes in the sample.

1 - phyper(0, 2, 98, 10)

###############################################################################

### Poisson probabilities

## Example 1:  Y ~ Pois(lambda = 2.6)

# What is P(Y = 4)?

dpois(4, lambda = 2.6)

# What is P(Y >= 2)?
# This is the probability of TWO OR MORE occurrences.

1 - ppois(1, lambda = 2.6)


# What is P(3 <= Y <= 6)?
# This is the probability of between 3 and 6 occurrences.

ppois(6, lambda = 2.6) - ppois(2, lambda = 2.6)

## (Partial) plot of this Poisson distribution:

# Y ~ Pois(lambda = 2.6):

plot(0:12, dpois(0:12, lambda = 2.6), type='h', lwd=2, xlab='y', ylab='p(y)')

###############################################################################