One reason statisticians recommend random samples and randomized experiments is so that they avoid bias by letting chance decide who gets selected or who receives which treatment. Something very important happens when we observe the results of coin tossing, random sampling, and random assignments: we find that chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run. This is the basis for the idea of probability.
The Law of Large Numbers says that if we observe more and more repetitions of any chance process, the proportion of times that a specific outcome occurs approaches a single value. We call this value the probability.
The probability is a number between zero and one that describes the proportion of times the outcome would occur in a very long series of repetitions. Probability gives us a language to describe the long-term regularity of random behavior.
Outcomes that never occur have a probability of 0. An outcome that happens on every repetition has a probability of 1. An outcome that happens half the time in a very long series of trials has a probability of 0.5. While we can never observe a probability exactly, the mathematics of probability is based on imagining what would happen in an indefinitely long series of trials.
We can imagine this using a simulation, an imitation of chance behavior based on a model that accurately reflects the situation.
When trying to solve any statistical problem, you should follow this 4-step process to make sure you answer all parts of the problem asked.
- State: What’s the question that you're trying to answer?
- Plan: How will you go about answering the question? What statistical techniques does this problem call for?
- Do: Make graphs and carry out needed calculations. Don’t forget labels and units!
- Conclude: Give your conclusion in the setting of the real-world problem. Units, units, units!
Probability Models
The description of a chance process has two parts:
- A list of possible outcomes (the sample space S)
- A probability for each outcome
The sample space S of a chance process is the set of all possible outcomes. The sample space for a coin toss would be written as S {Heads, Tails}.
A probability model is a description of some chance process that consists of two parts: a sample space S and a probability for each outcome.
An event in probability is any collection of outcomes from some chance process. That is, an event is a subset of the sample space. Events are usually designated by capital letters. If A is any event, we write its probability as P(A) and say “P of A.”