### Problem 1

Suppose 65% SML201 students like World Coffee better than Hoagie Haven. We selected a random sample of 20 SML201 students, and asked them which they prefer. What is the probability that more than 18 students said “World Coffee”? Write R code to compute the actual probability.

Hint:

• We solved a very similar problem in class on April 2 with coin tosses. Each student answer can be thought of as being like a coin toss. Decide whether you want “Heads” to represent HH or WC, and then solve the problem using pbinom, just like we did in class.

#### Solution

This is like asking about the probability of a coin’s coming up heads 18 times or more out of 20 when the probability of the coin’s coming up heads is 65%:

1 - pbinom(q = 18, size = 20, prob = 0.65)
## [1] 0.00213312

Here’s another way to compute the answer:

sum(dbinom(x = c(19, 20), size = 20, prob = 0.65))
## [1] 0.00213312

### Problem 2

In class, we saw several ways to compute the cumulative probability for the binomial distribution: we used pbinom; we summed up the outputs of dbinom; we also generated a large sample using rbinom, and then computed the proportion of the generated numbers that was under a certain threshold.

#### Part 2(a)

Write a function named MyPbinom1, which works just like pbinom. You may use dbinom but not rbinom in the function you write.

MyPbinom1 <- function(q, size, prob){
return(sum(dbinom(x = 0:q, size = size, prob = prob )))
}

MyPbinom1(q = 2, size = 10, prob = 0.45)
## [1] 0.09955965
pbinom(q = 2, size = 10, prob = 0.45)
## [1] 0.09955965

#### Part 2(b)

Write a function named MyPbinom2, which works just like pbinom. You may use rbinom but not dbinom in the function you write.

MyPbinom2 <- function(q, size, prob){
sample <- rbinom(n = 10000000, size = size, prob = prob)
return(mean(sample <= q))
}

MyPbinom2(q = 2, size = 10, prob = 0.45)
## [1] 0.0994306
pbinom(q = 2, size = 10, prob = 0.45)
## [1] 0.09955965