Chapter 10 - Introducing Probability Probability: The mathematics of chance behavior. Probability Vocab Random = individual outcomes are uncertain, but a large number of repetitions produces a regular distribution. Probability = The proportion of times an outcome would occur in a long series of repetitions. Probability Theory = The branch of mathematics that describes random behavior. Probability Factoids: Any probability is a number between 0 and 1. All possible outcomes together MUST have a total probability equaling 1. The probability that an event does NOT occur = 1- the probability that it DOES occur.

If 2 events have no outcomes in common, the probability that one OR the other occurs = the SUM of their individual probabilities… ex: Blood Type (Afr. American distribution) Type O A B AB Probability .49 .27 .20 ? What is the probability that the person chosen has either type A or type B blood? (.27 + .20) = .47 What is the probability of type AB blood? (.49 + .27 + .20) = .96 -----> (1 - .96) = .04

Probability Models Random Variable = A variable whose value is a numerical outcome of a random phenomenon. Probability Model/Distribution = Describes what the possible values of a variable are and the probabilities assigned to those values. 2 requirements: a) Every probability in the distribution must be a number between 0 and 1. b) The sum of the probabilities in the distribution must =1. Sample space (S) = The set of all possible outcomes. Event = An outcome or set of outcomes from the sample space. ex: Coin toss: Sample space is S = {H, T} / Event = Toss a head ex: Roll 2 Dice: Sample space (S) = Event = “Roll a 5” = (1, 4), (2, 3), (3, 2), (4, 1) = 4/36 outcomes =1/9 36 possible outcomes

Probability Model Types Finite: Fixed and limited number of outcomes. Continuous: Outcomes may take on any value in an interval of numbers. Finite Probability Model example: Student’s grade on a 4.0 scale (A = 4.0) X is the random variable representing the grade of a student chosen at random: Grade 0 1 2 3 4 Probability .10 .15 .30 .30 .15 Probabilities add to 1 The probability that a student got a B or better is expressed as: P(X > 3) = P(X = 3) + P(X = 4) = .30 + .15 = .45

P(X) = Area of shaded region Continuous Probability Model The probability dist. of X is described by a density curve. X is defined as an interval of values rather than just one value. The probability of any one single value = 0. P(X) = Area of the interval under the density curve. ex: P(X) = Area of shaded region

Normal Curves Normal distributions ARE probability distributions Heights of young women ex: X = height of random woman 18-25 yrs old (in inches) X has distribution N(64.5, 2.5) What is the prob. that a randomly chosen young woman is between 68 and 70 inches tall? P(68 < X < 70) =