DefinitionsA probability is a measure of the likelihood that an event in the future will happen. It can only assume a value between 0 and 1. A value near zero means the event is not likely to happen. A value near one means it is likely. There are three ways of assigning probability: ◦ classical, ◦ empirical, and ◦ subjective.
Basic Statements About Probability1. The probability, P, of any event or state of nature occurring is greater than or equal to 0 and less than or equal to 1. That is: 0 P(event) 12. The sum of the simple probabilities for all possible outcomes of an activity must equal 1.3. Probability „p‟ of the happening of an event is also known as probability of success & „q‟ the non-happening of the event as the probability of failure.4. If P(E) = 1, E is called a certain event & if P(E) = 0, E is called an impossible event
Simple Definitions Trial & Event ◦ Example: - Consider an experiment which, though repeated under essentially identical conditions, does not give unique results but may result in any one of the several possible outcomes. ◦ Experiment is known as a Trial & the outcomes are known as Events or Cases. Throwing a die is a Trial & getting 1 (2,3,…,6) is an event. Tossing a coin is a Trial & getting Head (H) or Tail (T) is an event.
A probability experiment is a chance process that leads to well-defined results called outcomes. An outcome is the result of a single trial of a probability experiment. A sample space is the set of all possible outcomes of a probability experiment. An event is the collection of one or more outcomes of an experiment
Assigning ProbabilitiesThree approaches to assigningprobabilities --◦Classical◦Empirical◦Subjective 7
Mathematical/ Classical/ „a priori‟ Probability Basic assumption of classical approach is that the outcomes of a random experiment are “equally likely”. According to Laplace, a French Mathematician: “Probability, is the ratio of the number of „favorable‟ cases to the total number of equally likely cases”. If the probability of occurrence of A is denoted by p(A), then by this definition, we have: 8
Limitations of Classical definition Classical probability is often called a priori probability because if one keeps using orderly examples of unbiased dice, fair coin, etc. one can state the answer in advance (a priori) without rolling a dice, tossing a coin etc. Classical definition of probability is not very satisfactory because of the following reasons: ◦ It fails when the number of possible outcomes of the experiment is infinite. ◦ It is based on the cases which are “equally likely” and as such cannot be applied to experiments where the outcomes are not equally likely.
Relative/ Statistical/ Empirical Probability Empirical Probability of an event is an "estimate" that the event will happen based on how often the event occurs after collecting data or running an experiment (in a large number of trials). It is based specifically on direct observations or experiences. Empirical Probability Formula P(E) = probability that an event, E, will occur. n(E) = number of equally likely outcomes of E. n(S) = number of equally likely outcomes of sample space S.
Limitations of Statistical/ Empiricalmethod The Empirical probability P(A) defined earlier can never be obtained in practice and we can only attempt at a close estimate of P(A) by making N sufficiently large. The experimental conditions may not remain essentially homogeneous and identical in a large number of repetitions of the experiment. The relative frequency of m/N, may not attain a unique value, no matter however large N may be.
Subjective Probability If there is little or no past experience or information on which to base a probability, it may be arrived at subjectively. Illustrations of subjective probability are: 1. Estimating the likelihood Tiger Woods will win the Grand Slam in 2009. 2. Estimating the likelihood you will become a millionaire by 2015. 3. Probability President Obama will win the 2012 Presidential election.
Glossary of terms Classical Probability: It is based on the idea that certain occurrences are equally likely. ◦ Example: - Numbers 1, 2, 3, 4, 5, & 6 on a fair die are each equally likely to occur. Conditional Probability: The probability that an event occurs given the outcome of some other event. Independent Events: Events are independent if the occurrence of one event does not affect the occurrence of another event. Joint Probability: Is the likelihood that 2 or more events will happen at the same time. Multiplication Formula: If there are m ways of doing one thing and n ways of doing another thing, there are m x n ways of doing both.
Outcome: Observation or measurement of an experiment. Prior Probability: The initial probability based on the present level of information. Probability: A value between 0 and 1, inclusive, describing the relative possibility (chance or likelihood) an event will occur. Subjective Probability: Synonym for personal probability. Involves personal judgment, information, intuition, & other subjective evaluation criteria. ◦ Example: - A physician assessing the probability of a patient‟s recovery is making a personal judgment based on what they know and feel about the situation.