Probability: The Mathematics of Chance Abdolaziz Abdollahi Affiliation: Shiraz University e-mail: abdolahi@shirazu.ac.ir Probability: The Mathematics of Chance Chapter 8 Sarah Cameron November 13, 2018

Basic Terms Randomness Probability A phenomenon is said to be random if the individual outcomes are uncertain but the long-term pattern of many individual outcomes is predictable. Random does not mean haphazard. Probability The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions.

Sample Space, Event, Probability Model The sample space S of a random phenomenon is the set of all possible outcomes. An event is any outcome or any set of outcomes of a random phenomenon. That is, an event is a subset of the sample space. A probability model is a mathematical description of a random phenomenon consisting of two parts: a sample space S and a way of assigning probabilities to events.

Probability Rules The probability P(A) of any event A satisfies 0 ≤ P(A) ≤ 1. If S is the sample space in a probability model, then P(S) = 1. Two events A and B are disjoint if they have no outcomes in common and so can never occur together. If A and B are disjoint, P(A or B) = P(A) + P(B) This is the addition rule for disjoint events. 4. The complement of any event A is the event that A does not occur, written as Ac. The complement rule states that P(Ac) = 1 – P(A)

Discrete Probability Model A probability model with a finite sample space. To assign probabilities in a discrete model, list the probability of all the individual outcomes. These probabilities must be numbers between 0 and 1, and their total must equal 1. The probability of any event is the sum of the probabilities of the outcomes making up the event.

Benford’s Law 1st Digit 1 2 3 4 5 6 7 8 9 Probability .301 .176 .125 .097 .079 .067 .058 .051 .046 Q: What is the probability that the first digit is an 8 or 9? A: .097 Q: What is the probability that the first digit is 1 through 4? A: .699

Equally Likely Outcomes If a random phenomenon has k possible outcomes, all equally likely, then each individual outcome has probability 1/k. The probability of any event A is P(A) = count of outcomes in A k

Density Curves and Continuous Probability Models A density curve is a curve that Is always on or above the horizontal axis. Has an area of exactly 1 underneath of it. A continuous probability model assigns probabilities as area under a density curve. The area under the curve and above any range of values is the probability of an outcome in that range.

The Mean and Standard Deviation of a Probability Model Suppose that the possible outcomes x1 , x2 , … , xk in a sample space S are numbers and that pj is the probability of outcome xj : The mean μ of this probability model is μ = x1p1 + x2p2 + … + xkpk The variance σ2 of this probability model is σ2 = (x1 – μ) 2p1 + (x2 – μ) 2p2 + … + (xk – μ) 2pk The standard deviation σ is the square root of the variance.

The Law of Large Numbers Observe any random phenomenon having numerical outcomes with finite mean μ. According to the law of large numbers, as the random phenomenon is repeated a large number of times, The proportion of trials on which each outcome occurs gets closer and closer to the probability of the outcome. The mean x of the observed values gets closer and closer to μ.

Central Limit Theorem Draw an SRS of size n from any large population with mean μ and finite standard deviation σ. Then: The mean of the sampling distribution of x is μ. The standard deviation of the sampling distribution of x is σ/√n. The sampling distribution of x is approximately normal when the sample size n is large.

Discussion & Homework Applications: Homework Problems: Working in a casino/gambling Catching fraud Homework Problems: (7th Edition): 13, 36