Sampling and Data: Probability 1.5 2008/03/31 15:01:05 GMT-5 2008/07/08 20:58:19.999 GMT-5 Barbara Illowsky cnx@cnx.org Susan Dean cnx@cnx.org Connexions cnx@cnx.org likelihood probability random randomness statistics This module introduces the concept of probability as a mathematical measure of randomness, including a number of real-world applications. Note: This module is currently under revision, and its content is subject to change. This module is being prepared as part of a statistics textbook that will be available for the Fall 2008 semester. Probability is the mathematical tool used to study randomness. It deals with the chance of an event occurring. For example, if you toss a fair coin 4 times, the outcomes may not be 2 heads and 2 tails. However, if you toss the same coin 4,000 times, the outcomes will be close to 2,000 heads and 2,000 tails. The expected theoretical probability of heads in any one toss is 1 2 or 0.5. Even though the outcomes of a few repetitions are uncertain, there is a regular pattern of outcomes when there are many repetitions. After reading about the English statistician Karl Pearson who tossed a coin 24,000 times with a result of 12,012 heads, one of the authors tossed a coin 2,000 times. The results were 996 heads. The fraction 996 2000 is equal to 0.498 which is very close to 0.5, the expected probability. The theory of probability began with the study of games of chance such as poker. Today, probability is used to predict the likelihood of an earthquake, of rain, or whether you will get a A in this course. Doctors use probability to determine the chance of a vaccination causing the disease the vaccination is suppose to prevent. A stockbroker uses probability to determine the rate of return on a client's investments. You might use probability to decide to buy a lottery ticket or not. In your study of statistics, you will use the power of mathematics through probability calculations to analyze and interpret your data. Probability A number between 0 and 1, inclusive, that gives the likelihood that a specific event will occur. More exact, the foundation of statistics are given by the following 3 axioms (by A. N. Kolmogorov, 1930’s): Let S denote the sample space, A and B are any two events in S . Then: (1). 0≤P(A)≤1; size 12{0 <= P \( A \) <= 1;} {} (2). If A and B are any two mutually exclusive events, then P ( A or B ) = P ( A ) + P ( B ) ; (3). P ( S ) = 1 .