Lab 2C - Which Song Plays Next?

Directions: Follow along with the slides, completing the questions in blue on your computer, and answering the questions in red in your journal.

A new direction

For the past two labs, we've looked at ways that we can summarize data with numbers.

– Specifically, you learned how to describe the center, shape and spread of variables in our data.
In this lab, we're going to estimate the probability that a rap song will be chosen from a playlist with both rap and rock songs, if the choice is made at random.

– The playlist we'll work with has 100 songs: 39 are rap and 61 are rock.

Estimate what ... ?

To estimate the probability, we're going to imagine that we select a song at random, write down its genre (rock or rap), put the song back in the playlist, and repeat 499 more times for a total of 500 times.
The statistical question we want to address is: On average, what proportion of our selections will be rap?
Why do we put a song back each time we make a selection?
What would happen in our little experiment if we did not do this?

Calculating probabilities

Remember that a probability is the long-run proportion of time an event occurs.

– Many probabilities can be answered exactly with just a little math.

– The probability we draw a single rap song from our playlist of 39 rap and 61 rock songs is 39/100, 0.39 or 39%.
Probabilities can also be answered exactly if we were willing to randomly select a song from the playlist, write down its genre, place the song back in the list, and repeatedly do this forever.

– Literally, forever ...

– But we don't have that much time. So we're only going to do it 500 times which will give us an estimate of the probability.

Estimating probabilities

You might ask, Why are we estimating the probability if we know the answer is 39%?

– Sometimes, probabilities are too hard to calculate with simple division as we did above. In which case, we can often program a computer to run an experiment to estimate the probability.

– We refer to these programs as simulations.
The techniques you learn in this lab could be applied to very simple probability calculations or very hard and complex calculations.

– In both cases, your estimated probability would be very close to the actual probability.

Getting ready

Simulations are meant to mimic what happens in real-life using randomness and computers.

– Before we can start simulating picking songs from a playlist, we need to simulate that playlist in R.
Simulate our 39 rap songs using the repeat rep() function.
```
rap <- rep("rap", times = 39)
```
Look in the Environment pane for the vector containing your rap songs.
Use a similar line of code to simulate the rock songs in our playlist of 100.

Put the songs in the playlist

Now that we've got some different songs, we need to combine them together.

– To do this, we can use the combine function c() in R.
Fill in the blanks to combine your different songs:
```
songs <- __(rap, ____)
```
And with that, our playlist of songs should be ready to go.

– Type songs into the console and hit enter to see your individual songs.

Pick a song, any song

Data scientists call the act of choosing things randomly from a set, sampling.

– We can randomly choose a song from our playlist by using:
```
sample(songs, size = 1, replace = TRUE)
```
Run this code 10 times and compute the proportion of "rap" songs you drew from the 10.
- Vocabulary Check: A proportion is a fraction of the whole.
  - For example, if 2 rap songs were drawn from the 10, the proportion would be 2/10
  - It is more common to express a proportion as a decimal, in this case, 0.20
  - It is even more common to express a proportion as a percentage, 20%
Once everyone in your class has computed their proportions, calculate the range of proportions (the largest proportion minus the smallest proportion) for your class and write it down.

Now do() it some more

Instead of running the same line of code multiple times ourselves we can use R to do() multiple repetitions for us.

– Fill in the blanks below to do the sample code from the previous slide 50 times:
```
do(___) * sample(___, ___ = ___, ___ = ___)
```
Recall that we need to store our results to be able to perform analysis.
Assign the 50 selected songs the name draws and then View your file.
What is the variable name?
- R defaulted to naming the variable based on the function used. You may use the data cleaning skills you learned in lab 6 to rename the variable if you wish.
Fill in the blank below to tally how often each genre was selected:
```
tally(~___, data = draws)
```
Compute the proportion of "rap" songs for your 50 draws and find out if the range for your class' proportions is bigger or smaller than when we drew 10 songs.

Proportions vs. Probability

To review, so far in this lab we've:

– Simulated a "playlist" of songs.

– Repeatedly simulated drawing a song from the playlist, noting its genre and placing it back in the playlist.

– Computed the proportion of the draws that were "rap".
These proportions are all estimates of the theoretical probability of choosing a rap song from a playlist.

– As we increase the number of draws, the range of proportions should shrink.

When using simulations to estimate probabilities, using a large number of repeats is better because the estimates have less variability and so we can be confident we're closer to the actual value.

Non-random Randomness

We've seen that random simulations can produce many different outcomes.

– Some estimated probabilities in your class were smaller/larger relative to others.
There are instances where you might like the same random events to occur for everyone.

– We can do this by using set.seed().

For example, the output of this code will always be the same:

set.seed(123)
sample(songs, size = 1, replace = TRUE)

## [1] "rap"

Playing with seeds

With a partner, choose a number to include in set.seed then redo the simulation of 50 songs.

– Both partners should run set.seed(___) just before simulating the 50 draws.

– The blank in set.seed(___) should be the same number for both partners.

– Verify that both partners compute the same proportion of "rap" songs.
Redo the 50 simulations one last time but have each partner choose a different number for set.seed(___).

– Are the proportions still the same? If so, can you find two different values for set.seed that give different answers?

On your own

Suppose there are 1,200 students at your school. 400 of them went to the movies last Friday, 600 went to the park and the rest read at home.

If we select a student at random, what is the probability that this student is one of those who went to the movies last Friday?

Answer this by estimating the probability that a randomly chosen student went to the movies using 500 simulations.

– Write down both the estimated probability and the code you used to compute your estimate. You might find it helpful to write your answer in an R Script (File -> New File -> R Script)

– Include set.seed(123) in your code before you do 500 repeated samples.