# Example code for finding binomial probabilities in R

# The command qbinom computes _cumulative_ binomial probabilities 

# To get the cumulative probability, the syntax is: qbinom(x, n, p) 
# where p, n, and x are values that you fill in 

# Suppose our random variable X is binomial with n=20 and p=0.65 

# Let's answer the following questions 
# 1. What is the probability of 13 or fewer successes?  P(X <= 13) 

pbinom(13, 20, 0.65)

# 2. What is the probability of 6 or fewer successes? P(X <= 6)  

pbinom(6, 20, 0.65)

# 3. What is the probability of EXACTLY 13 successes? P(X = 13) 

pbinom(13, 20, 0.65) - pbinom(12, 20, 0.65)

# or more simply:

dbinom(13, 20, 0.65)

# 4. What is the probability of more than 13 successes? P(X > 13) 

1 - pbinom(13, 20, 0.65)

# 5. What is the probability of 13 or more successes? P(X >= 13) 

1 - pbinom(12, 20, 0.65)

# 6. What is the probability of 8, 9, or 10 successes? P(8 <= X <= 10) 

pbinom(10, 20, 0.65) - pbinom(7, 20, 0.65)


# Notice how the commands mirror the examples we did in class using the tables 

# The difference is:  R can do these for any n and p, whereas the table only 
# gives probabilities for certain n and p choices.