Probability

Probability Definitions and Relationships Sample space: All the possible outcomes that can occur. Simple event: one outcome in the sample space; a possible outcome of a random circumstance. Event: a collection of one or more simple events in the sample space; often written as A, B, C, and so on.

Assigning Probabilities A probability is a value between 0 and 1 and is written either as a fraction or as a proportion. A probability simply is a number between 0 and 1 that is assigned to a possible outcome of a random circumstance. For the complete set of distinct possible outcomes of a random circumstance, the total of the assigned probabilities must equal 1.

Estimating Probabilities from Observed Categorical Data - Empirical Approach Assuming data are representative, the probability of a particular outcome is estimated to be the relative frequency (proportion) with which that outcome was observed.

Nightlights and Myopia Assuming these data are representative of a larger population, what is the approximate probability that someone from that population who sleeps with a nightlight in early childhood will develop some degree of myopia? Note: = 79 of the 232 nightlight users developed some degree of myopia. So we estimate the probability to be 79/232 = 0.34.

Complementary Events Note: P(A) + P( A C ) = 1 One event is the complement of another event if the two events do not contain any of the same simple events and together they cover the entire sample space. Notation: A C represents the complement of A. Example: A Simple Lottery (cont) A = player buying single ticket wins A C = player does not win P(A) = 1/1000 so P(A C ) = 999/1000

Mutually Exclusive Events Two events are mutually exclusive if they do not contain any of the same simple events (outcomes). Example; A Simple Lottery A = all three digits are the same. B = the first and last digits are different The events A and B are mutually exclusive.

Independent and Dependent Events Two events are independent of each other if knowing that one will occur (or has occurred) does not change the probability that the other occurs. Two events are dependent if knowing that one will occur (or has occurred) changes the probability that the other occurs.

Example Independent Events Customers put business card in restaurant glass bowl. Drawing held once a week for free lunch. You and Vanessa put a card in two consecutive wks. Event A = You win in week 1. Event B = Vanessa wins in week 2 Events A and B refer to to different random circumstances and are independent.

Event A = Alicia is selected to answer Question 1. Event B = Alicia is selected to answer Question 2. P(A) = 1/50. If event A occurs, her name is no longer in the bag; P(B) = 0. If event A does not occur, there are 49 names in the bag (including Alicia’s name), so P(B) = 1/49. Events A and B refer to different random circumstances, but are A and B independent events? Knowing whether A occurred changes P(B). Thus, the events A and B are not independent. Example: Dependent Events

Joint and Marginal Probabilities These probabilities refer to the proportion of an event as a fraction of the total.

Unions and intersections P{A  B}  P{A} + P{B} because A and B do overlap. P{A  B} = P{A} + P{B} - P{A  B}. A  B is the intersection of A and B; it includes everything that is in both A and B, and is counted twice if we add P{A} and P{B}.

Conditional Probability Consider two events A and B. What is the probability of A, given the information that B occurred? P(A | B) = ? Example: –What is the probability that a women is married given that she is years old?

Probability Problems P(Married | 18-29) = 7842/ 22,512

Conditional probability and independence If we know that one event has occurred it may change our view of the probability of another event. Let –A = {rain today}, B = {rain tomorrow}, C = {rain in 90 days time} It is likely that knowledge that A has occurred will change your view of the probability that B will occur, but not of the probability that C will occur. We write P(B|A)  P(B), P(C|A) = P(C). P(B|A) denotes the conditional probability of B, given A. We say that A and C are independent, but A and B are not. Note that for independent events P(A  C) = P(A)P(C).

Conditional probability - tornado forecasting Consider the classic data set on the next Slide consisting of forecasts and observations of tornados (Finley, 1884). Let –F = {Tornado forecast} –T = {Tornado observed} Use the frequencies in the table to estimate probabilities – it’s a large sample, so estimates should not be too bad.

Forecasts of tornados

Conditional probability - tornado forecasting P(T) = 51/2803 = P(T  F) = 28/2803 P(T|F) = 28/100 = P(T|F c ) = 23/2703 = –Knowledge of the forecast changes P(T). F and T are not independent. P(F|T) = 28/51 = –P(T|F), P(F|T) are often confused but are different quantities, and can take very different values.

Continuous and discrete random variables A continuous random variable is one which can (in theory) take any value in some range, for example crop yield, maximum temperature. A discrete variable has a countable set of values. They may be –counts, such as numbers of accidents –categories, such as much above average, above average, near average, below average, much below average –binary variables, such as dropout/no dropout

Probability distributions If we measure a random variable many times, we can build up a distribution of the values it can take. Imagine an underlying distribution of values which we would get if it was possible to take more and more measurements under the same conditions. This gives the probability distribution for the variable.

Continuous probability distributions Because continuous random variables can take all values in a range, it is not possible to assign probabilities to individual values. Instead we have a continuous curve, called a probability density function, which allows us to calculate the probability a value within any interval. This probability is calculated as the area under the curve between the values of interest. The total area under the curve must equal 1.

Normal (Gaussian) distributions Normal (also known as Gaussian) distributions are by far the most commonly used family of continuous distributions. They are ‘bell-shaped’ –and are indexed by two parameters: –The mean  – the distribution is symmetric about this value –The standard deviation  – this determines the spread of the distribution. Roughly 2/3 of the distribution lies within 1 standard deviation of the mean, and 95% within 2 standard deviations.

The probability of continuous variables IQ test –Mean = 100 and sd = 15 What is the probability of randomly selecting an individual with a test score of 130 or greater? –P(X ≤ 95)? –P(X ≥ 112)? –P(X ≤ 95 or X ≥ 112)?

The probability of continuous variables (cont.) What is the probability of randomly selecting three people with a test score greater than 112? –Remember the multiplication rule for independent events.