
# 4: Probability


• 4.1: Terminology
In this module we learned the basic terminology of probability. The set of all possible outcomes of an experiment is called the sample space. Events are subsets of the sample space, and they are assigned a probability that is a number between zero and one, inclusive.
• 4.2: Independent and Mutually Exclusive Events
Two events A and B are independent if the knowledge that one occurred does not affect the chance the other occurs. If they are not independent, then they are dependent. In sampling with replacement, with selecting each member with the possibility of being chosen more than once, and the events are considered to be independent. In sampling without replacement, each member may be chosen only once, and the events are considered not to be independent. When events do not share outcomes, they are mutu
• 4.3: Two Basic Rules of Probability
The multiplication rule and the addition rule are used for computing the probability of A and B, and the probability of A or B for two given events A, B. In sampling with replacement each member has the possibility of being chosen more than once, and the events are considered to be independent. In sampling without replacement, each member may be chosen only once, and the events are not independent. The events A and B are mutually exclusive events when they have no common outcomes.
• 4.4: Binomial Distribution
A statistical experiment can be classified as a binomial experiment if the following conditions are met: (1) There are a fixed number of trials. (2)There are only two possible outcomes: "success" or "failure" for each trial. (3) The trials are independent and are repeated using identical conditions. The outcomes of a binomial experiment fit a binomial probability distribution.
• 4.5: Mean or Expected Value and Standard Deviation
The expected value is often referred to as the "long-term" average or mean. This means that over the long term of doing an experiment over and over, you would expect this average. This “long-term average” is known as the mean or expected value of the experiment and is denoted by the Greek letter μμ . In other words, after conducting many trials of an experiment, you would expect this average value.
• 4.6: The Standard Normal Distribution
A z-score is a standardized value. Its distribution is the standard normal, Z∼N(0,1). The mean of the z-scores is zero and the standard deviation is one. If y is the z-score for a value x from the normal distribution N(μ,σ) then z tells you how many standard deviations x is above (greater than) or below (less than) μ.
• 4.7: Using the Normal Distribution
The normal distribution, which is continuous, is the most important of all the probability distributions. Its graph is bell-shaped. This bell-shaped curve is used in almost all disciplines. Since it is a continuous distribution, the total area under the curve is one. The parameters of the normal are the mean μ and the standard deviation σ. A special normal distribution, called the standard normal distribution is the distribution of z-scores. Its mean is zero, and its standard deviation is one.