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
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 are 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
BUSINESS
MATHEMATICS
&
STATISTICS