1	Chapter 6 Section A Fundamentals of Probability
2	Terminology An event is a result of an experiment or an observation of some phenomenon. The probability of an event is a mathematical quantity describing the likelihood that the event happens. Probabilities range from 0 to 1. If an event is CERTAIN, the probability is 1. If an event is an impossibility its probability is 0.
3	Terminology An theoretical probability is one based on assumptions about the event in question. An empirical probability is one based on the actual results of observations or experiments. A subjective estimate of a probability is one made through personal judgment or intuition.
4	Example An theoretical probability is one based on assumptions about the event in question. If a standard 6-sided die is rolled, the probability of rolling: an even number is 1/2. P(E) = 1/2 a 5 is 1/6. P(5) = 1/6 a number greater than 4 is 1/3. P(>4) = 1/3
5	Theoretical Probability If the outcomes are equally likely: Count the total number of possible outcomes of an event. Count the number of outcomes that represent success - that is, the number of outcomes that represent the desired result. Determine the probability of success by dividing the number of successes by the total number of possible outcomes.
6	Theoretical Probability If the outcomes are equally likely,
7	Theoretical Probability Probability of winning the lottery, where winning involves picking 6 numbers from 1-52. (order irrelevant, no repetitions)
8	Theoretical Probability Given 5 cards from a standard 52 card deck, find the probability that all 5 are spades.
9	Empirical Probability During 1997, it was observed that it rained on 115 different days. For a given day, what is the probability that it rained?
10	Probability Distributions For a given experiment, the probability is 1 that SOMETHING happens. Therefore, if we were to add the probabilities for all possible events we should get 1.
11	Probability Distributions Consider flipping two coins and counting the number of heads on any toss. We should have P(0) + P(1) + P(2) = 1. P(0) = 1/4 P(1) = 1/2 P(2) = 1/4 1/4 + 1/2 + 1/4 = 1
12	Why those probabilities?
13	Empirical Techniques Empirical techniques are used when theoretical techniques cannot be used. For example, if you are interested in the probability of an engine part failing, there is no theoretical way to calculate that probability.
14	Empirical Techniques You would then take some number of randomly selected parts and test them, counting the number that failed and arriving at your probability.
15	Intuition Suppose a fair coin has been flipped 9 times and has come up heads each time. What is the probability it will come up heads again? 1/2 The coin is not smart enough to know what it has been doing.
16	Intuition If a region has flooded 4 times in the past 2000 years such a flood is called a “500 year flood.” Is it possible for a region to have a 500 year flood 2 years in a row? Yes!
17	Intuition Is it likely a region will have a 500 year flood 2 years in a row? No. What is the probability of it happening? Wait till next section to find out how to calculate such a probability.
18	Probability of an Even NOT Happening
19	Probability of an Even Not Happening