BUSINESS MATHEMATICS STATISTICS LECTURE 38 Patterns of probability

BUSINESS MATHEMATICS & STATISTICS

LECTURE 38 Patterns of probability: Binomial, Poisson and Normal Distributions Part 2

CUMULATIVE BINOMIAL PROBABILITIES Probability of r or more successes in n trials with the probability of success in each trial 1. Look in column for n 2. Look in column for r 3. Look at column for value of p(0. 05 to 0. 5) Example n = 5; r = 4; p = 0. 5 p( 4 or more successes in 5 trials) = 0. 1874 = 18. 74 %

BINOMDIST Returns the individual term binomial distribution probability Use BINOMDIST in problems with a fixed number of tests or trials, when the outcomes of any trial are only success or failure, when trials are independent, and when the probability of success is constant throughout the experiment. For example, BINOMDIST can calculate the probability that two of the next three babies born are male.

BINOMDIST(number_s, trials, probability_s, cumulative) Number_s is the number of successes in trials. Trials is the number of independent trials. Probability_s is the probability of success on each trial. Cumulative is a logical value that determines the form of the function. If cumulative is TRUE, then BINOMDIST returns the cumulative distribution function, which is the probability that there at most number_s successes; if FALSE, it returns the probability mass function, which is the probability that there are number_s successes.

EXAMPLE USING TABLES p(2 or less dry days) + p(3 or more dry days) = 1 p(2 or less dry days) = 1 - p(3 or more dry days) n = 7; r = 3; p = 0. 4 p(3 or more dry days) = 0. 5800 p(2 or less dry days) = 1 - 0. 5800 = 0. 4200 = Chance of 5 or more wet days next week

EXAMPLE 8 bit message is transmitted electronically p(one bit transmitted erroneously) = 0. 1 What is the chance that entire message is transmitted correctly)? n = 8; r = 8, p = 0. 1; p(exactly 0 errors)? p(0 errors) + p( 1 or more errors) = 1 P(0 errors) = 1 – p( 1 or more errors) TABLES = 1 – 0. 5695 = 0. 4305

EXAMPLE A surgery is successful for 75% patients p(success in at leaset 7 cases in randomly selected 9 patients)? n = 9; p(success) = 0. 75; p(at lease 7 cases)? p = 0. 75 is outside the table p(failure) = 1 – 0. 75 = 0. 25 Success at least 7 = Failure 2 or less P(failure 2 or less) = 1 – p(failure 3 or more) = 1 – 0. 3993 = 0. 6007 = 60%

NEGBINOMDIST Returns the negative binomial distribution NEGBINOMDIST returns the probability that there will be number_f failures before the number_s-th success, when the constant probability of a success is probability_s This function is similar to the binomial distribution, except that the number of successes is fixed, and the number of trials is variable Like the binomial, trials are assumed to be independent.

NEGBINOMDIST Example You need to find 10 people with excellent reflexes, and you know the probability that a candidate has these qualifications is 0. 3 NEGBINOMDIST calculates the probability that you will interview a certain number of unqualified candidates before finding all 10 qualified candidates.

NEGBINOMDIST Syntax NEGBINOMDIST(number_f, number_s, probability_s) Number_f is the number of failures. Number_s is the threshold number of successes. Probability_s is the probability of a success.

CRITBINOM Returns the smallest value for which the cumulative binomial distribution is greater than or equal to a criterion value Use this function for quality assurance applications For example, use CRITBINOM to determine the greatest number of defective parts that are allowed to come off an assembly line run without rejecting the entire lot. Syntax CRITBINOM(trials, probability_s, alpha) Trials is the number of Bernoulli trials. Probability_s is the probability of a success on each trial. Alpha is the criterion value.

BUSINESS MATHEMATICS & STATISTICS