Probability Probability theory is the branch of mathematics that studies chances and the long run patterns of randomness A phenomenon is called random if individual outcomes are unpredictable but the long term pattern of many individual outcomes is predictable The probability of any outcome of a random phenomenon is the fraction of times the outcome would occur in a very long series of repetitions A probabilistic model is a mathematical description of a random phenomenon consisting of two parts a sample space S and a way of assigning probabilities to events For example if we toss a coin it is impossible to predict in advance whether the outcome will be a head or tail Our intuition tells us that it is equally likely to be a head or a tail and somehow we sense that if we repeat the experiment of tossing a coin a large number of times head will occur about half the time otherwise the coin is considered to be unfair In other words the long term pattern of this experiment should be predictable To check this out we recently flipped a coin 1 000 times and obtained 460 heads 460 and 540 tails The percentage of heads is 1 000 0 46 46 which is called the relative frequency If an experiment is repeated n times and an event occurs m times then m is called the relative frequency of the event n Our task is to create a model that will assign a number p between 0 and 1 called the probability of an event which will predict the relative frequency This means that for a sufficiently large number of repetitions of an experiment we should have p m n Probabilities can be obtained in one of the following ways 1 Empirical probabilities also called experimental probabilities are obtained from experimental data For example an assembly line producing brake assemblies for GM produces 1 500 items per day The probability of producing a defective brake can be obtained by experimentation Suppose the 1 500 brakes are tested and 3 are found to be defective Then the empirical probability is the relative frequency of occurrence namely 3 0 002 or 0 2 1 500 2 Subjective probabilities are obtained by experience and indicate a measure of certainty on the part of the speaker These probabilities are not necessary arrived at through experimentation or computation For example a TV meteorologist studies the satellite map and then issues a prediction about tomorrow s weather based on experience 80 of rain tomorrow 3 Theoretical probabilities are obtained by logical reasoning according to stated definitions Our focus is on theoretical probabilities but we should keep in mind that our theoretical model should be predictive of the results obtained by experimentation empirical probabilities if they are not consistent and we have been careful about record keeping in arriving at an empirical probability we should conclude that our model is faulty On the other hand if our model has been proved to be correct then we would have the following The Law of Averages also called the law of large numbers Consider an experiment in which the theoretical probability of an event A is p Suppose that the single trial of this experiment is repeated many times and that the outcome of each trial is independent of the others If the number of trials increases the empirical probability of A will approach the theoretical value p Terminology An experiment E is an observation of any physical occurrence The sample space S of an experiment is the set of all its possible outcomes An event A is a subset of the sample space If an event can never occur its probability is defined to be 0 If an event is definitely going to occur then its probability is defined to be 1 Definition of Theoretical Probability If an experiment E has n E possible outcomes and each outcome is equally likely to occur and if n A of these outcomes are in an event A then the probability of the event A denoted by P A is defined to be number of outcomes in event A P A n A number of all possible outcomes n E Conclusion the probability P A of an event A is a number between 0 and 1 the bigger this number the more likely this event is going to occur Mutually Exclusive Events Two events A and B of the same experiment E are said to be mutually exclusive if they have nothing in common In other words A and B will not happen at the same time For instance let E be the experiment of picking a card randomly from a standard deck of 52 cards let A be the event of picking a Diamond and let B be the event of picking a Spade Then clearly they are disjoint because we cannot find a card that is both a Diamond and a Spade If A and B are mutually exclusive events of the same experiment E then P A or B P A P B Complementary Events Two events A and B of the same experiment E are said to be complementary if either one of them but not both must happen in any single run of the experiment E More precisely A and B are complementary if A B empty set and A B Sample space of E Examples Let E be the experiment of randomly picking a whole number bigger than 1 Let A be the event of picking a prime number and B be the event of picking a composite number Then the two events A and B are complementary because any whole number bigger than 1 must be either prime or composite but not both Let E be the experiment of throwing a dart towards a dartboard Let A be the event of hitting the bull s eye and B be the event of missing the bull s eye Then the two events A and B are complementary Let E be the experiment of picking 3 balls from a jar containing some red balls yellow balls and green balls Let A be the event of picking up some red balls and let B be the event of picking up no red ball then A and B are complementary events Probability of Complementary events If A and B are complementary events in the same experiment E then P A P B 1 Consequently if we know P A we can easily compute P B Examples 1 Suppose that we are given a bias coin that the probability of flipping a head is 0 4 then the probability of flipping a tail is 1 0 4 0 6 In the language of percentages we have Prob flipping a tail 100 Prob flipping a head 100 40 60 2 In a local sweepstake the probability of winning the grand prize is 0 01 Then the probability of not winning the grand prize is 100 0 01 99 99 3 A jar contains 3 red balls 3 yellow balls and 3 green balls If 3 balls are taken out randomly and sequentially without replacement what …