Recall from last lecture we studied

• Random sampling: where we select the sample from the population in an unbiased fashion
• Bias: A sample is biased if some observational units have a higher probability of being included in the sample.

• Keywords: random and probability

• In short: Probability is the study of randomness.
• Its roots lie in one historical constant.

In 79 AD a volcanic eruption in the Roman city of Pompeii covered the entire city in hot ash, perfectly preserving many items…

… including what I saw at a Pompeii exhibit in Montreal:

There are two approaches to studying probability:

Mathematically (MATH 310)	Simulations

• Note: The "random simulation" in question is not limited to coin flips.

• The mathematical approach to studying probability requires more mathematical background: set theory, discrete math, calculus, and later on measure theory.
• The simulation approach does not.
• Instead we require a computer's random number generator to generate simulations. Why?

Doing this repeatedly by hand is tiring:

• Analogy for rest of course: POWERBALL
• 
• We can model any random simulation by tinkering with the following attributes of Powerball:

• Atrributes of the Lottery Machine:
  • How many balls do you have?
  • What are written on the balls?
  • Do the balls have equal probability of being picked?
• Attributes of the Drawing:
  • How are you drawing the balls?
  • How many balls do you draw?
  • What are you recording about each drawn ball?
  • What do you do with drawn balls?
• Number of Lotteries:
  • How many times do you repeat the lottery?

All hail the mosaic package, which is pre-installed on RStudio Server. Load as usual: library(mosaic)

The following 4 functions will give us (most of) the random simulation tools we need:

1. rflip(): Flip a coin
2. shuffle(): Shuffle a set of values (akin to cards)
3. do(): Do the same thing many, many, many times
4. resample(): the swiss army knife of functions