Business Analytics Chapter 4

Probability

The numerical measure of the likelihood that an event will occur.

Random experiment

A process that generates well-defined outcomes.

Outcome

A possible result of a random experiment.

Sample space

The set of all possible outcomes of a random experiment.

Event

A collection of outcomes.

Probability of an event

The sum of the probabilities of the outcomes belonging to the event.

Complement of an event

The event consisting of all outcomes that are not in the original event.

Complement rule

P(Aᶜ) = 1 − P(A).

Union of two events

The event containing all outcomes that are in A or B or both.

Union notation

A ∪ B.

Intersection of two events

The event containing all outcomes that are in both A and B.

Intersection notation

A ∩ B.

Addition law

P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

Why is the intersection subtracted in the addition law?

Because outcomes in the intersection are counted twice when P(A) and P(B) are added.

Mutually exclusive events

Events that have no outcomes in common and therefore cannot occur together.

Mutually exclusive condition

P(A ∩ B) = 0.

Addition law for mutually exclusive events

P(A ∪ B) = P(A) + P(B).

Joint probability

The probability of the intersection of two events.

Joint probability table

A table showing probabilities for combinations of two events.

Marginal probability

The probability of a single event, found by summing the appropriate row or column of a joint probability table.

Conditional probability

The probability of one event given that another event has occurred.

Conditional probability notation

P(B|A) means the probability of B given that A has occurred.

Conditional probability formula

P(B|A) = P(A ∩ B) / P(A).

Independent events

Events where knowing that one event occurred does not change the probability of the other event.

Independent event condition

P(B|A) = P(B) or P(A|B) = P(A).

Dependent events

Events where the occurrence of one event changes the probability of the other event.

Mutually exclusive vs. independent

Two events with nonzero probabilities cannot be both mutually exclusive and independent.

Multiplication law

P(A ∩ B) = P(A)P(B|A) or P(B)P(A|B).

Multiplication law for independent events

P(A ∩ B) = P(A)P(B).

Bayes' theorem

A method for calculating revised or posterior probabilities using prior probabilities and conditional probabilities.

Prior probability

The initial probability before new information is considered.

Posterior probability

The revised probability after new information is considered.

Bayes' theorem purpose

To update a probability when new information becomes available.

Probability tree

A diagram that shows prior probabilities followed by conditional probabilities and their possible outcomes.

Random variable

A numerical description of an experimental outcome.

Discrete random variable

A random variable that can assume either a finite number of values or an infinite sequence of values.

Continuous random variable

A random variable that can assume any numerical value within an interval or collection of intervals.

Discrete vs. continuous random variable

Discrete variables have countable possible values; continuous variables can take any value within an interval.

Probability distribution

A description of the range and relative likelihood of possible values for a random variable.

Probability mass function (PMF)

The function that provides the probability for each value of a discrete random variable.

Requirements for a valid discrete probability distribution

Each probability must be between 0 and 1, and all probabilities must sum to 1.

Empirical probability distribution

A probability distribution generated from observations.

Relative frequency method

A method for generating probabilities in an empirical probability distribution using observed frequencies.

Expected value

The mean or weighted average of a random variable, where the probabilities are the weights.

Expected value formula

E(X) = ΣxP(x).

SUMPRODUCT

Excel function that can be used to calculate the expected value of a discrete random variable.

Variance of a discrete random variable

A measure of variability calculated as the weighted average of squared deviations from the mean.

Variance formula

Var(X) = Σ(x − μ)²P(x).

Standard deviation of a random variable

The positive square root of the variance.

Standard deviation formula

σ = √Var(X).

Discrete uniform probability distribution

A discrete probability distribution in which all possible values are equally likely.

Discrete uniform distribution example

A fair die, where each of the six outcomes is equally likely.

Binomial probability distribution

A distribution used to determine the probability of a specified number of successes in a fixed number of independent trials when there are only two possible outcomes.

Binomial requirements

Fixed number of trials, two possible outcomes, independent trials, and constant probability of success.

Binomial variables

n = number of trials; x = number of successes; p = probability of success; 1 − p = probability of failure.

Binomial probability formula

P(X = x) = C(n,x)pˣ(1−p)ⁿ⁻ˣ.

Binomial distribution example

Number of customers who place an order out of a fixed number of customers contacted.

Poisson probability distribution

A distribution used to model the number of occurrences over a specified interval of time or space.

Poisson requirements

The probability of an occurrence is the same for equal-length intervals, and occurrences in separate intervals are independent.

Poisson parameter λ

The expected value or mean number of occurrences in the specified interval.

Poisson probability formula

P(X = x) = e⁻λ λˣ / x!.

Poisson example

Number of patients arriving at a hospital during a specified time interval.

Changing the interval for a Poisson distribution

If the time or space interval changes, the expected number of occurrences must be adjusted to match the new interval.

Continuous probability distribution

A distribution for a continuous random variable where probabilities are calculated over intervals.

Probability density function (PDF)

The function used with continuous random variables; it does not directly give the probability of a particular value.

Probability of a particular value for a continuous random variable

The probability of any particular exact value is zero.

Continuous probability calculation

For continuous variables, probabilities are calculated over intervals using area under the probability density function.

Uniform probability distribution

A continuous distribution where probability is proportional to the length of the interval.

Uniform distribution parameters

a = smallest possible value and b = largest possible value.

Uniform distribution expected value

μ = (a + b) / 2.

Uniform distribution variance

σ² = (b − a)² / 12.

Uniform distribution probability

P(a ≤ X ≤ b) is based on the proportion of the interval's length.

Triangular probability distribution

A continuous distribution used when the minimum, maximum, and most likely value (mode) are known.

Triangular distribution parameters

a = minimum, b = maximum, and m = most likely value.

Normal probability distribution

A continuous probability distribution with a bell-shaped curve.

Normal distribution parameters

The mean (μ) and standard deviation (σ).

Normal distribution shape

The normal distribution is symmetric and bell-shaped.

Normal distribution mean

The mean is also the median and mode and is the highest point of the normal curve.

Normal distribution symmetry

The area to the left of the mean equals 0.5 and the area to the right equals 0.5.

Normal distribution standard deviation

The standard deviation determines how wide and flat the normal curve is.

Larger normal standard deviation

A larger standard deviation produces a wider and flatter normal curve.

Empirical rule for normal distribution

Approximately 68% of values are within 1 standard deviation, 95% within 2 standard deviations, and almost all within 3 standard deviations of the mean.

Normal probability using area

Probability for a range of values is represented by the area under the normal curve over that interval.

Finding a normal value from a probability

Use the specified probability to determine the corresponding value of the normally distributed random variable.

Exponential probability distribution

A continuous distribution used to model the length of time or distance between occurrences.

Exponential distribution purpose

Used for the interval between occurrences, such as the time between business loan defaults.

Exponential distribution mean

The mean represents the expected length of the interval between occurrences.

Exponential cumulative probability

Used to calculate the probability that the interval is less than or equal to a specified value.

Risk and uncertainty

Probability distributions help businesses evaluate possible outcomes and make decisions involving uncertainty.