Probability

Sample Space

• An experiment is a sequence of operations carried out under specified conditions.
• The result of an experiment is called an outcome.
• If an experiment's outcomes are not predetermined, then the experiment is called a chance experiment.
• definition. The sample space, denoted ${\Omega,}$ is the set of all possible outcomes of a statistical experiment. Each outcome in a sample space is called a sample point.
• definition. A probability model is a sample space and an operation of assigning probabilities.
• example. When we toss a coin, we have a sample space with two outcomes: $$\Omega = \set{H, T}.$$ where ${H}$ is heads and ${T}$ is tails.
• example. If we toss the coin three times, then the sample space is: $$\Omega = \left\lbrace \begin{matrix} HHH & HHT \\ HTH & THH \\ HTT & TTH \\ THT & TTT \end{matrix} \right\rbrace.$$
  • The sample space resulting from tossing a coin finitely many times is an example of a discrete and finite sample space.
• example. Consider an experiment where we roll a tetrahedral (4-sided) die.
  • We can visualize the sample space with a grid:

    

    If we roll a ${2}$ on the first roll and a ${3}$ on the second, we would have the outcome:

    

  • Alternatively, we can visualize the sample space with a tree:

    

• example. Suppose a rifle fires at the square target below. We mark the point where the bullet hits. Suppose further that the shooter always hits the target with infinite precision. In this case, the sample space is infinite and continuous.

  

Probability Axioms

Events

• Suppose a rifleman shoots at the target below.

  

  What is the probability that the center is hit with infinite precision?
  • It's impossible to hit the center with infinite precision. No matter how close we get, there's always a coordinate ${(x,y)}$ that's closer. The probability of hitting the center with infinite precision is ${0.}$
  • For this reason, we assign probabilities to events rather than outcomes.

    

• definition. Given a sample space ${\Omega,}$ and event ${E}$ is a subset of ${\Omega.}$