# 1. Probability and Two-Way Tables Intro

For this sub-competency you will be introduced to the basics of probability.

#### Basic Probability Rules

Probability will play a huge role later in this course when we start investigating the probability of obtaining certain results from a sample. An unusual event is one that has a low probability of occurring. This is not a precise definition, because how low is “low?” Typically, probabilities of 5% or less are considered low. Recall that 5% means 5 per 100 or 5 times out of 100. Therefore, an event E with a 5% chance of occurring means that in repeated trials we would expect to see E happen in only 5 trials out of every 100.  Thus, events with probabilities of 5% or lower are considered unusual. However, this cutoff point can (and will) vary by the context of the problem.

Probability is basically the science of chance behavior. Chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run. This is why we will use probability to gain useful results from random samples and randomized comparative experiments…although we don’t know exactly what we’ll see from our sampling or experimentation, if we repeat the process over and over, we gain some confidence in the outcomes we’ll see.

Here are some definitions you want to be familiar with. An experiment is a repeatable process where the results are uncertain. An outcome is one specific possible result from the experiment. The set of all possible outcomes is the sample space.

##### Example

A basketball player shoots three free throws. What are the possible sequences of hits (H) and misses (M)? The experiment in this case is a basketball player shooting 3 free throws. A possible outcome of this experiment is the sequence HHM (hit, hit, miss). The sample space of this experiment is:

S  =  {HHH, HHM, HMH, HMM, MHH, MHM, MMH, MMM}

Note that there are 8 outcomes in this sample space, as each free through has 2 possibilities (hit or miss). So 2 ⋅ 2 ⋅ 2 = 23 = 8. You can often create a sample space using a graphical approach, as shown: 