Basic Probability Lecture 9

Lab 1 What’s in a problem? What’s in a research question? Hypotheses Operationalizing Concepts

Probability Are the things that we observe different from what would be expected by chance? ‘Chance’ turns out to be a tough concept, but for now we limit ourselves to things that can be repeated, independently under same conditions.

A human probability problem Myers and Chen (1996) conducted a study where they tested teenagers who had previously participated in an experiment 12 years earlier. The teenagers were presented with objects, only one of which they had seen 12 years ago. Suppose we present them with one object they had seen and one they had not seen. How do we determine if their choice responses are ‘real’ or just chance? Basic probability problem with real data

Such probability problems require two things: Probability Distribution What would the distribution be like if it were only due to chance? Decision Rule What criteria do we need in order to determine whether an observation is just due to chance or not. These principles are essential to “inferential statistics”, or using samples to make inferences about populations. All statistics and sampling statistics are, at their core, based on probability. Thus, we start with some key concepts of probability that will serve us the rest of the semester.

Probability Distributions The probability of an outcome is the relative frequency that an event can be expected to occur. The probability distribution is the set of relative frequencies for every possible outcome. If our sample was just 4 students, would 3 out of 4 be enough? What is going to happen when we increase our sample size? Probability distribution of number of correct responses for 4 teenagers when p(correct) = .5

Probability Concepts Basic Concepts in Probability Basic Probability Rules Special Types of Probability Joint Probabilities Probabilities of Unions of Events Conditional Probabilities

Basic Concepts in Elementary Probability Random Selection Every possibility has equal chance of being chosen. Independence The probability of a response on one trial does not depend on the outcome of any other trials. Elementary Event Possible outcomes of a probability experiment E.g., each coin toss Sample Space The complete set of elementary events E.g., all coin tosses

Mutually exclusive, exhaustive, events Mutually exclusive events Two or more events that cannot occur at the same time. Exhaustive events A set of events that accounts for all of the elementary events in the sample space.

Basic rules of probability Multiplication Rule For independent events, we can multiply the probabilities together to get the probability for all of the events occurring. Example: Probability of rolling a die twice and getting 6 on both rolls. But what happens if the events are not independent? Example: probability of selecting a club from a deck of cards, then selecting another club (without replacement)?

Multiplication Rule when two or more events will happen at the same time, and the events are independent, then the special rule of multiplication law is used to find the joint probability: P(X and Y) = P(X) x P(Y) when two or more events will happen at the same time, and the events are dependent, then the general rule of multiplication law is used to find the joint probability: P(X and Y) = P(X) x P(Y|X) Dice are easy example: rolling 2 dice and getting a pair of 6’s? (1/36) Cards are favorite example: Probability of drawing 2 aces? 4/52 * 3/51 = 1/221

Basic rules of probability (continued) The addition rule For independent events, we can add the probabilities to get the probability of either event occurring. Example: Rolling die and getting a 4 or a 6. Again, what happens if the events are not independent (in this case, mutually exclusive)?

Addition Rule When two or more events will happen at the same time, and the events are mutually exclusive, then: P(X or Y) = P(X) + P(Y) When two or more events will happen at the same time, and the events are not mutually exclusive, then: P(X or Y) = P(X) + P(Y) - P(X and Y) For example, what is the probability that a card chosen at random from a deck of cards will either be a king or a heart? P(King or Heart) = P(X or Y) = 4/52 + 13/52 - 1/52 = 30.77% Remember: multiplication rule is for “X and Y” while addition rule is for “X or Y”

Special Types of Probability Joint Probabilities Probabilities of Unions of Events Conditional Probabilities

Joint Probabilities Probability of obtaining a particular combination of events. E.g., probability of flipping a coin twice and getting heads both times. Just use multiplication rule! P (A and B) = n(A and B) / n (S) What about non-independent events? E.g., probability of a given respondent in a survey being female and voting for reproductive rights?

Union Probabilities A union of two elementary events consists of all the elementary events belonging to either of them. Examples: Probability of flipping a coin and it being heads or tails. (mutually exclusive union) Non-independent events: (pres_data.dta) Probability of being a female or rating the president with a 4. p(E1) + p(E2) – p(E1 and E2) (.48) + (.16) – (.06) = .58 Pres_data allows us to look at a different example– probabilities based on empirical data.

Conditional Probability Probability of an event occurring given that another event has occurred. Example: probability of rating the president as 4 (agree) given that the respondent is a female. Example: pres_data.dta P(E1|E2) = 6/48 = .125 3 Doors Problem Remember: Joint probability is probability of selecting two or more things from all possibilities; conditional probability is probability of selecting something given something else (e.g., limiting the sample)

Fun with random numbers: Drawing sample from a normal distribution… Probability links to sampling in several ways– the first is just in how we connect our sample to the population from which it was drawn.

The Probability Problem and Sampling Any given sample may not be representative of the population, even if randomly drawn. Non-representative samples will lead to erroneous statements about the population However…basic probability theory can helps us determine what the population probably looks like based on our sample.

Potential Problems in a Sample Obtaining a representative sample 1930’s Literary Digest poll: Franklin Roosevelt predicted to lose the 1936 presidential election by a landslide. Oops… he won by a landslide. Accounting for outliers in sample What is the best undergraduate major if you want a high income (UNC-Chapel Hill survey)? One outlier, a geography major named Michael Jordan, accounted for the huge skew in average salaries for graduates (at the time he made $80 million/year)