What Do You Expect?

Law of Large Numbers

The law states, in effect, that as more trials of an experiment are conducted, the experimental probability more closely approximates the theoretical probability. It is not at all unusual to have 100% heads after three tosses of a fair coin, but it would be extremely unusual to have even 60% heads after 1000 tosses.

Payoff

The number of points (or dollars or other objects of value) a player in a game recieves for a particular outcome.

Binomial Probability

The probability of getting one of 2 possible outcomes over many trials. For example, the probability of getting a heads or tails when tossing a coin or the probability of getting a 5 or not 5 when rolling a number cube.

Relative Frequency

The ratio of the number of desired results to the total number to trials.

Sample Space

The set of all possible outcomes in a probability situation. When you toss two coins, the sample space consists of four outcomes: HH, HT, TH, and TT.

Equally Likely

Two or more events that have the same probability of occuring. Example: when you toss a fair coin, heads and tails are equally likely. Each has a 50% chance of happening. Rolling a six-sided dice gives a 1/6 probability for each number to come up. Each outcome is equally likely.

Area Model

A diagram in which fractions of the area of the diagram correspond to probabilities in a situation. For example, suppose there are three blue blocks and 2 red blocks in a container. If 1 block is drawn out at a time, and the block drawn each time is replaced, the area model below shows that the probability of getting 2 red blocks is 4/25. Area models are particularly helpful when the outcomes being analyzed are not equally likely, because more likely outcomes take up larger areas. Area models are also helpful for outcomes involving more than one stage, such as rolling a number cube and then tossing a coin.

Tree Diagram

A diagram used to determine the number of possible outcomes in a probability situation. The number of final branches is equal to the number of possible outcomes.

Fair Game

A game in which each player is equally likely to win. The probability of winning a two person fair game is 1/2. An unfair game can be made fair by adjusting the scoring system or the payoffs. Example: suppose you play a game in which two fair coins are tossed. You score when both coins land heads up Otherwise, your opponent scores. The probability that you will score is 1/4 and the probability that your opponent will score is 3/4. To make the game fair, you might adjust the scoring system so that you receive 2 points each time you score and your opponent receives 1.

Probability

A number between 0 and 1 that describes the likelihood that an outcome will occur.

Outcome

A possible result. When determining probabilities, it is important to be clear about what the possible outcomes are.

Theoretical Probability

A probability obtained by analyzing a situation. If all the outcomes are equally likely, you can find a theoretical probability of an event by listing all the possible outcomes and then finding the ration of the number of outcome producing the desired event to the total number of outcomes.

Experiemental Probability

A probability that is determined through experimentation. Example: You could find the experimental probability of getting a head when you toss a coin by tossing a coin many times and keeping track of the outcomes. The experimental probability would be the ratio of the number of heads to the total number of tosses or trials. Experimental probability may not be the same as the the theoretical probability . However, for a large number of trials, the are likely to be close. Experimental probabilities are used to predict behavior over the long run.

Compound Event

An event that consists of two or more simple events. Example: tossing a coin is a simple event. Tossing two coins and examining combinations of outcomes, is a compound event.

Simulation

An experiment using objects that represent the relevant characteristics of a real-world situation.

Favorable Outcome

An outcome that gives a desired result. A favorable outcome is sometimes called a success. Example: when you toss two coins to find the probability of the coins matching, HH and TT are favorable outcomes.

Expected Value (long-term average)

Intuitively, the average payoff over the long-run. Example: suppose you are playing a game with two number cubes. You score 2 points when the sum of 6 is rolled. 1 point for the sum of three, and 0 points for anything else. If you roll the number cubes 36 times, you could expect to roll the sum of six five times, the sum of three twice, and the other sums 29 times. This means that you could expect to score (5x2)+(2x1)+(29x0)=12 points for 36 rolls, and average of 12/36=1/3 point per roll. Here 1/3 is the expected value or long-term average of 1 roll.

Trial

One round of an experiment.

Random

Outcomes that are uncertain when viewed individually, but which exhibit a predictable patter over many trials.