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.

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`

The answer is 0.2%.

Here’s another way to compute the answer:

`sum(dbinom(x = c(19, 20), size = 20, prob = 0.65))`

`## [1] 0.00213312`

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.

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`

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`