Introduction to Probability

• PROBABILITY - is a numerical way to express the likelihood of an event happening.

  It's like a score assigned to how probable an event is.
• The Probability Scale (0-1):

  0 = Impossible Event:

  The event has no chance of happening (e.g., flipping heads and tails on the same coin toss).

  1 = Certain Event:

  The event is guaranteed to happen (e.g., the sun will rise tomorrow)
• FORMULA:

  Probability: Favorable Outcomes/Total Outcomes
• EXPERIMENTS - Any process that generates well-defined outcomes.
• SAMPLE SPACE - The set of all sample points (experimental outcomes).

  - Often denoted by S
• Possible Events - Events that can occur, but that cannot be predicted.
• The two main rules for assigning probabilities:

  • Between 0 and 1:

  Each outcome's probability (let's call it P(E)) has to be between 0 and 1. Imagine a scale from 0 (impossible) to 1 (guaranteed). The probability sits somewhere on this scale.

  • Sum equals 1:

  The sum of ALL the probabilities for every outcome in the experiment must equal 1. This makes sense because all the outcomes together represent every possibility, so their combined probability is 1 (certain to happen)
• ACCEPTABLE METHODS FOR ASSIGNING PROBABILITIES:

  • Classical Methods
  • Relative Frequency Methods
  • Subjective Method
• • Classical Methods:

  This assumes all outcomes are equally likely (like rolling a fair die or coin tossing). In this case, probability is calculated by the number of favorable outcomes divided by the total number of outcomes

  Example: Coin Toss = Number of favorable outcomes//Total possible number of outcomes
• • Relative Frequency Methods:

  This method involves performing the experiment many times and observing how often each outcome occurs. The probability is then estimated based on the observed frequency of each outcome.

  Example: Not a fair die roll = Number of times you observed 3/ Total number of rolls
• • Subjective Method:

  This method relies on personal judgment and experience to assign probabilities. It's less exact but can be useful when actual experiments are difficult or impossible.