Probability and Counting Rules

Probability"Probability theory is nothing but common sense reduced to calculation." - P.S. Laplaceprobability"Probability theory is nothing but common sense reduced to calculation." - P.S. Laplaceprobability"Probability theory is nothing but common sense reduced to calculation." - P.S. Laplaceprobability"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceProbabilityExampleA card is drawn from an ordinary deck of card. Find these probabilities Getting king of heartsGetting a spadeGetting a 5 or a clubsGetting a 5 or a 7Getting a card which is not a spadeGetting 14 of clubs"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceProbability"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceProbability"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceProbability"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceProbability"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceProbability"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceProbability"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceProbability"Probability theory is nothing but common sense reduced to calculation." - P.S. Laplaceprobability"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceProbabilityExampleIn a sample of 50 college students, 18 are freshmen, 23 are sophomore, 2 are junior, and 7 are senior. Set up a frequency distribution and find the following probabilities: A student is a freshmanA student is a freshman or a sophomoreA student is neither a freshman nor a juniorA student is not a seniorSolution

"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceYear LevelFrequencyFreshman18Sophomore23Junior2Senior7Total50ProbabilityExampleIn a sample of 50 college students, 18 are freshmen, 23 are sophomore, 2 are junior, and 7 are senior. Set up a frequency distribution and find the following probabilities: A student is a freshmanA student is a freshman or a sophomoreA student is neither a freshman nor a juniorA student is not a senior"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceProbabilityExampleIn a sample of 50 college students, 18 are freshmen, 23 are sophomore, 2 are junior, and 7 are senior. Set up a frequency distribution and find the following probabilities: A student is a freshmanA student is a freshman or a sophomoreA student is neither a freshman nor a juniorA student is not a senior"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceProbabilityExampleIn a sample of 50 college students, 18 are freshmen, 23 are sophomore, 2 are junior, and 7 are senior. Set up a frequency distribution and find the following probabilities: A student is a freshmanA student is a freshman or a sophomoreA student is neither a freshman nor a juniorA student is not a senior"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceProbabilityExampleIn a sample of 50 college students, 18 are freshmen, 23 are sophomore, 2 are junior, and 7 are senior. Set up a frequency distribution and find the following probabilities: A student is a freshmanA student is a freshman or a sophomoreA student is neither a freshman nor a juniorA student is not a senior"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceprobabilityThree Conceptual Approaches to ProbabilitySubjective ProbabilitySubjective probability is the probability assigned to an event based on subjective judgment, experience, information, and belief. For example,A sportswriter may say that there is 90% probability that University of the East Red Warriors will win the UAAP championships.A physician may say that, on the basis of his diagnosis, there is a 60% chance that the patient will recover.A financial analysis may say that there is 80% probability that peso dollar exchange rate will decrease by 3 pesos."Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceThe addition rule and multiplication ruleProbability"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceAddition Rule and Multiplication RuleA lot of problems involve determine the probability of two or more events. This is when independent, dependent, and mutually exclusive comes into the picture in dealing with probability.There are important things to note about mutually exclusive, independent, and dependent events.The first is of which is mutually exclusive are always dependentSecondly is independent events are never mutually exclusiveLastly is dependent events may or may not be mutually exclusive.The exception of the first and second is that when at least one of the two events has a zero probability."Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceAddition Rule and Multiplication Rule"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceAddition Rule and Multiplication Rule"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceAddition Rule and Multiplication Rule"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceAddition Rule and Multiplication Rule"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceAddition Rule and Multiplication Rule"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceAddition Rule and Multiplication RuleExampleIn a certain insurance company there are 20 senior salespersons and 30 junior salespersons; 8 senior and 14 junior salespersons are males. If a salesperson is selected, find the probability that the salesperson is a senior or a female.SolutionThe sample space

"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceSalespersonMaleFemaleTotalSenior181220Junior141630Total222850Addition Rule and Multiplication RuleExampleIn a certain insurance company there are 20 senior salespersons and 30 junior salespersons; 8 senior and 14 junior salespersons are males. If a salesperson is selected, find the probability that the salesperson is a senior or a female."Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceAddition Rule and Multiplication Rule"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceAddition Rule and Multiplication Rule"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceAddition Rule and Multiplication Rule"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceAddition Rule and Multiplication Rule"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceAddition Rule and Multiplication Rule"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceAddition Rule and Multiplication Rule"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceAddition Rule and Multiplication Rule"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceAddition Rule and Multiplication Rule"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceMarginal and Conditional ProbabilitiesProbability"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceMarginal and Conditional Probabilities"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceMarginal and Conditional Probabilities"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceMarginal and Conditional Probabilities"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceMarginal and Conditional Probabilities"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceRandom Variables and Discrete Probability DistributionProbability"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceRandom Variables and Discrete Probability DistributionSuppose the above table shows the frequency and relative frequency distribution of the number of TV owned by 500 families residing in the City of Manila."Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceRandom Variables and Discrete Probability DistributionSuppose one family is randomly selected from this population. The process of random selecting a family is called a random or chance experiment.Let X denote the number of TVs owned by the selected family. Then X can assume any of the 4 possible values (0, 1, 2, and 3) recorded in the leftmost column of the table. The value assumed by X depends on the family is selected.Hence, this value depends on the outcome of a random experiment. Therefore, X is referred to random variable."Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceRandom Variables and Discrete Probability DistributionA random variable is a function or rule that assigns a number to each outcome of an experiment, it is called chance variable.In general, a random variable is denoted by X.A random variable can be discrete or continuous.A discrete random variable assumes values that can be counted, whileA continuous random variable can assume all values between any two specific values;A variable obtained by measuring, or Contained one or more intervals"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceRandom Variables and Discrete Probability Distribution"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceRandom Variables and Discrete Probability Distribution"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceRandom Variables and Discrete Probability DistributionExampleConstruct a probability distribution for tossing three coins. Let X represent the number of tails.SolutionThe 8 possible events, and the corresponding values for X, are:"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceRandom Variables and Discrete Probability DistributionExampleConstruct a probability distribution for tossing three coins. Let X represent the number of tails.SolutionTherefore, the probability distribution for the number of tails occurring in three coin tosses is: "Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceBinomial probability distributionProbability"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceBinomial Probability DistributionA binomial experiment is one that possesses the following properties:The experiment consists of n repeated trials;Each trial results in an outcome that may be classified as a success or a failure (hence the name, binomial);The probability of a success, denoted by p, remains constant from trial to trial and repeated trials are independent.The number of successesXinntrials of a binomial experiment is called abinomial random variable."Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceBinomial Probability DistributionExamples of binomial experimentsTossing a coin 20 times to see how many tails occur.Asking 200 people if they watch ABC news.Rolling a die to see if a 5 appears.Examples which aren't binomial experimentsRolling a die until a 6 appears (not a fixed number of trials)Asking 20 people how old they are (not two outcomes)Drawing 5 cards from a deck for a poker hand (done without replacement, so not independent)"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceBinomial Probability Distribution"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceBinomial Probability DistributionExampleWhat is the probability of rolling exactly two sixes in 6 rolls of a die?SolutionAnytime a six appears, it is a success (denoted S) and anytime something else appears, it is a failure (denoted F). The ways you can get exactly 2 successes in 6 trials are given below. The probability of each is written to the right of the way it could occur. Because the trials are independent, the probability of the event (all six dice) is the product of each probability of each outcome (die):FFFFSS 5/6 * 5/6 * 5/6 * 5/6 * 1/6 * 1/6 = (1/6)^2 * (5/6)^4FFFSFS 5/6 * 5/6 * 5/6 * 1/6 * 5/6 * 1/6 = (1/6)^2 * (5/6)^4FFFSSF 5/6 * 5/6 * 5/6 * 1/6 * 1/6 * 5/6 = (1/6)^2 * (5/6)^4FFSFFS 5/6 * 5/6 * 1/6 * 5/6 * 5/6 * 1/6 = (1/6)^2 * (5/6)^4FFSFSF 5/6 * 5/6 * 1/6 * 5/6 * 1/6 * 5/6 = (1/6)^2 * (5/6)^4FFSSFF 5/6 * 5/6 * 1/6 * 1/6 * 5/6 * 5/6 = (1/6)^2 * (5/6)^4FSFFFS 5/6 * 1/6 * 5/6 * 5/6 * 5/6 * 1/6 = (1/6)^2 * (5/6)^4"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceBinomial Probability DistributionExampleWhat is the probability of rolling exactly two sixes in 6 rolls of a die?SolutionAnytime a six appears, it is a success (denoted S) and anytime something else appears, it is a failure (denoted F). The ways you can get exactly 2 successes in 6 trials are given below. The probability of each is written to the right of the way it could occur. Because the trials are independent, the probability of the event (all six dice) is the product of each probability of each outcome (die):FSFFSF 5/6 * 1/6 * 5/6 * 5/6 * 1/6 * 5/6 = (1/6)^2 * (5/6)^4FSFSFF 5/6 * 1/6 * 5/6 * 1/6 * 5/6 * 5/6 = (1/6)^2 * (5/6)^4FSSFFF 5/6 * 1/6 * 1/6 * 5/6 * 5/6 * 5/6 = (1/6)^2 * (5/6)^4SFFFFS 1/6 * 5/6 * 5/6 * 5/6 * 5/6 * 1/6 = (1/6)^2 * (5/6)^4SFFFSF 1/6 * 5/6 * 5/6 * 5/6 * 1/6 * 5/6 = (1/6)^2 * (5/6)^4SFFSFF 1/6 * 5/6 * 5/6 * 1/6 * 5/6 * 5/6 = (1/6)^2 * (5/6)^4SFSFFF 1/6 * 5/6 * 1/6 * 5/6 * 5/6 * 5/6 = (1/6)^2 * (5/6)^4SSFFFF 1/6 * 1/6 * 5/6 * 5/6 * 5/6 * 5/6 = (1/6)^2 * (5/6)^4"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceBinomial Probability DistributionExampleWhat is the probability of rolling exactly two sixes in 6 rolls of a die?SolutionNotice that each of the 15 probabilities are exactly the same: (1/6)^2 * (5/6)^4.Also, note that the 1/6 is the probability of success and you needed 2 successes. The 5/6 is the probability of failure, and if 2 of the 6 trials were success, then 4 of the 6 must be failures. Note that 2 is the value of x and 4 is the value of n-x.Further note that there are fifteen ways this can occur. This is the number of ways 2 successes can be occur in 6 trials without repetition and order not being important, or a combination of 6 things, 2 at a time."Probability theory is nothing but common sense reduced to calculation." - P.S. LaplaceBinomial Probability Distribution"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplacePoisson probability distributionProbability"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplacePoisson probability distributionA Poisson distribution is the probability distribution that results from a Poisson experiment.Attributes of a Poisson ExperimentAPoisson experimentis astatistical experimentthat has the following properties:The experiment results in outcomes that can be classified as successes or failures.The average number of successes () that occurs in a specified region is known.The probability that a success will occur is proportional to the size of the region.The probability that a success will occur in an extremely small region is virtually zero.Note that the specified region could take many forms. For instance, it could be a length, an area, a volume, a period of time, etc."Probability theory is nothing but common sense reduced to calculation." - P.S. LaplacePoisson probability distribution"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplacePoisson probability distributionPoisson DistributionAPoisson random variableis the number of successes that result from a Poisson experiment.The probability distributionof a Poisson random variable is called aPoisson distribution.The Poisson distribution has the following properties:The mean of the distribution is equal to .Thevarianceis also equal to ."Probability theory is nothing but common sense reduced to calculation." - P.S. LaplacePoisson probability distribution"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplacePoisson probability distribution"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplacePoisson probability distribution"Probability theory is nothing but common sense reduced to calculation." - P.S. LaplacePoisson probability distribution"Probability theory is nothing but common sense reduced to calculation." - P.S. Laplace