probability math

Theoretical Probability

what we expect to happen in theory.

Example.

When flipping a coin several times, half should be Heads and half Tails.

Dependent Events

the outcome of one event affects the outcome of the second.

Example.

drawing colored balls from a bag, when you do NOT put them back after each draw P(A and B) = P(A) × P(B|A)

Probability of 0

an impossible outcome

Example.

P(7) = 0 because it is impossible to roll a 7 on a single standard six-sided die

Probability

the likelihood of an event occurring. Compares the successful outcomes to the total outcomes.

Example.

Probability = number of successful outcomes possible / total number of outcomes possible

The probability of rolling a 4 is 1/6 ("one out of 6")

Placebo

something that looks like the actual treatment but designed to have no effect on the subject

Independent Events

two or more events that have no influence on each other.

Example.

drawing colored balls from a bag, when you put them back after each draw P(A and B) = P(A) × P(B)

Bias

when incorrect conclusions are made because some parts of the population are over-represented in the sample and some parts are under-represented

Population (in stats)

all members of a defined group

Example.

Every doctor in America

Experimental Investigation

Researchers assign subjects in the sample to certain treatments, then observe the effects of the treatment. Can show causation (cause and effect).

Example.

Does using algebra tiles during instruction help freshman students learn how to solve equations?

Experimental Probability / Empirical Probability

what actually occurs during a simulation or trial

Example.

When flipping a coin a few times, more are Heads than Tails.

Geometric Probability

interpreting one part of an area as a “successful” or “favorable” outcome, compared with the entire area

Margin of Error

a number that represents how far above or below the actual mean may be from the experimental mean in a study

Conditional Probability / Compound Event

occurs when two or more simple events are performed together so that one event happens and so does another

Sample

a subset of the population

Example.

The 250 doctors polled in a survey

Probability of 1

a guaranteed outcome

Example.

P (a number less than 10) = 1 because any roll on a standard six-sided die will guarantee a number smaller than 10

Observational Study

researchers observe and record data for the subjects in the sample (they do NOT assign treatments). Can only show correlation.

Example.

What percent of students at this university are involved in at least one extracurricular activity?

Probability Model

a mathematical representation of a random phenomenon

Sample Space

a list or set of all possible outcomes

Example.

tree diagram

Control Group

a group of specimens in an experiment to which no change is made; does not receive the experimental treatment

Example.

In an experiment investigating the effect of talk therapy on alleviating depression, the group receiving only the medicine would be the control group.

Simple Random Sample / Random Sample

a sample in which each member of the subset has an equal probability of being chosen

Example.

Every member is assigned a number and a random number generator is used to select the sample.

Experimental Group

a group of specimens in an experiment to which a change is made; receives the experimental treatment

Example.

In an experiment investigating the effect of talk therapy on alleviating depression, the group receiving the medicine and the therapy would be the experimental group.

Simulation

imitation of a real-world process or system

Example.

using a flight simulator to train pilots