Probability and Probability Distributions: Study Notes for Business Statistics

Introduction to Probability

Definition and Basics

Probability is the proportion of times an event is expected to occur when an experiment is repeated a large number of times under identical conditions. The probability of an event E is denoted as P(E), and it must satisfy:

• 0 ≤ P(E) ≤ 1

• P(E) = 0: Event never occurs

• P(E) = 1: Event always occurs

Example: The probability of flipping a head on a fair coin is 0.5.

Mutually Exclusive Events

• Two events are mutually exclusive if they cannot occur at the same time.

• For mutually exclusive events E1 and E2: P(E1 and E2) = 0

• The sum of probabilities of all mutually exclusive outcomes in an experiment is 1.

Example: In a coin toss, getting heads and tails are mutually exclusive.

Complementary Events

• The complement of event E is the event that E does not occur.

• P(E) + P(Ec) = 1

• P(E) = 1 - P(Ec)

Example: If the probability of rain is 0.3, the probability of no rain is 0.7.

Independent Events

• Events A and B are independent if the occurrence of one does not affect the probability of the other.

• P(A and B) = P(A) × P(B)

Example: Flipping a coin twice: the result of the first flip does not affect the second.

General Addition Rule

• For any two events E1 and E2:

$P(E_1 \text{ or } E_2) = P(E_1) + P(E_2) - P(E_1 \text{ and } E_2)$

Example: Probability of drawing a red card or an ace from a deck:

• P(red) = 26/52

• P(ace) = 4/52

• P(red ace) = 2/52

• P(red or ace) = 26/52 + 4/52 - 2/52 = 28/52

Sample Space and Counting Outcomes

• The sample space is the set of all possible outcomes.

• For independent events, total outcomes = product of outcomes for each event.

Example: Flipping a coin three times: 2 × 2 × 2 = 8 possible outcomes.

Classical Probability Formula

$P(E) = \frac{\text{Number of ways E can occur}}{\text{Total number of possible outcomes}}$

Example: Probability of getting exactly two heads in three coin flips: 3/8.

Relative Frequency Assessment

• Probability estimated by the proportion of times an event occurs in repeated trials.

$P(E) = \frac{\text{Number of times E occurs}}{n}$

Example: If 1570 out of 2250 Starbucks sales are caffeinated drinks, estimated probability = 1570/2250 ≈ 0.698.

Subjective Probability

• Probability based on personal judgment or belief, not on data.

Example: A fan estimates a 70% chance their team will win based on conviction.

Conditional Probability

• The probability of event A given that event B has occurred:

$P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$ (if $P(B) > 0$)

Example: Probability a card is the ace of diamonds given it is red: 1/26.

Dependent Events

• Events where the occurrence of one affects the probability of the other.

• For dependent events:

$P(A \text{ and } B) = P(A) \times P(B|A)$

Summary Table: Types of Events

Type	Definition	Key Formula
Mutually Exclusive	Cannot occur together	$P(A \text{ and } B) = 0$
Independent	Occurrence of one does not affect the other	$P(A \text{ and } B) = P(A)P(B)$
Dependent	Occurrence of one affects the other	$P(A \text{ and } B) = P(A)P(B|A)$

Discrete Probability Distributions

Random Variables

• Discrete random variable: Can take on a countable number of values (e.g., 0, 1, 2, ...).

• Continuous random variable: Can take on any value within an interval.

Probability Distribution Table

For a discrete random variable X, the probability distribution lists each possible value of X and its probability P(X).

X	P(X)
0	0.125
1	0.375
2	0.375
3	0.125

Example: Number of heads in three coin flips.

Expected Value (Mean) and Variance

• Expected value (mean): $E(X) = \sum x_i P(x_i)$

• Variance: $\sigma^2 = \sum (x_i - E(X))^2 P(x_i)$

• Standard deviation: $\sigma = \sqrt{\sigma^2}$

Example: For the above table, $E(X) = 1.5$, $\sigma^2 = 0.75$, $\sigma = 0.866$

Binomial Distribution

• Describes the probability of X successes in n independent trials, each with probability p of success.

• Conditions:

  • n identical trials

  • Each trial has two outcomes (success/failure)

  • Trials are independent

  • Probability of success (p) is constant

• Probability mass function:

$P(X = x) = \frac{n!}{x!(n-x)!} p^x q^{n-x}$, where $q = 1 - p$

Example: Probability of getting exactly 2 heads in 3 coin flips (p = 0.5): $P(X=2) = 3 \times 0.5^2 \times 0.5^1 = 0.375$

Mean and Variance of Binomial

• Mean: $E(X) = np$

• Variance: $\sigma^2 = npq$

Poisson Distribution

• Models the number of events occurring in a fixed interval of time or space.

• Parameter: $\lambda$ (average rate of occurrence)

• Probability mass function:

$P(X = x) = \frac{(\lambda t)^x e^{-\lambda t}}{x!}$

• Mean and variance: $E(X) = \lambda t$, $\sigma^2 = \lambda t$

Example: If a bank expects 16 customers per hour, the probability of 12 customers in 1 hour: $P(X=12) = \frac{16^{12} e^{-16}}{12!}$

Hypergeometric Distribution

• Used when sampling without replacement from a finite population.

• Parameters:

  • N: population size

  • K: number of successes in population

  • n: sample size

  • k: number of successes in sample

• Probability mass function:

$P(X = k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}}$

Example: Probability of drawing 2 red cards in 5 draws from a deck of 52 cards without replacement.

Continuous Probability Distributions

Normal Distribution

• Most important continuous distribution; bell-shaped, symmetric, unimodal.

• Defined by mean ($\mu$) and standard deviation ($\sigma$).

• Probability density function:

$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$

• Empirical Rule:

  • 68% within 1 standard deviation of mean

  • 95% within 2 standard deviations

  • 99% within 3 standard deviations

Standard Normal Distribution and Z-scores

• Standard normal: mean 0, standard deviation 1.

• Z-score formula:

$Z = \frac{X - \mu}{\sigma}$

• Use Z-tables or Excel to find probabilities.

Example: If $\mu = 100$, $\sigma = 20$, $X = 130$, $Z = (130-100)/20 = 1.5$

Finding Probabilities for Normal Distribution

• To find $P(X > a)$, first find $P(X < a)$, then $1 - P(X < a)$.

• To find $P(a < X < b)$, compute $P(X < b) - P(X < a)$.

Uniform Distribution

• All outcomes in an interval [a, b] are equally likely.

• Probability density function:

$f(x) = \begin{cases} \frac{1}{b-a} & a \leq x \leq b \\ 0 & \text{otherwise} \end{cases}$

• Mean: $\mu = \frac{a + b}{2}$

• Variance: $\sigma^2 = \frac{(b-a)^2}{12}$

Example: If tree growth is uniformly distributed between 1 and 4 inches per year, $f(x) = 1/3$ for $1 \leq x \leq 4$.

Exponential Distribution

• Models the time between events in a Poisson process.

• Probability density function:

$f(x) = \lambda e^{-\lambda x}$, for $x \geq 0$

• Mean: $1/\lambda$

• Variance: $1/\lambda^2$

Example: If customers arrive at a rate of 15 per 20 minutes, $\lambda = 15/20 = 0.75$ per minute. The probability the next customer arrives within 3 minutes: $P(X < 3) = 1 - e^{-0.75 \times 3}$

Summary Table: Choosing a Probability Distribution

Distribution	Data Type	Independence	Example
Binomial	Binary (success/failure)	Independent trials	Coin flips, pass/fail tests
Poisson	Count data	Independent events	Number of arrivals per hour
Hypergeometric	Binary or count	Dependent (no replacement)	Drawing cards without replacement
Normal	Continuous	—	Heights, test scores
Uniform	Continuous	—	Random number between a and b
Exponential	Continuous (time between events)	—	Time between arrivals

Applications and Examples

• Business: Estimating probability of sales, customer arrivals, or product defects.

• Finance: Modeling returns, risk, and rare events.

• Operations: Inventory management, queuing, and service times.

Key Excel Functions for Probability

• BINOM.DIST: Binomial probabilities

• POISSON.DIST: Poisson probabilities

• HYPGEOM.DIST: Hypergeometric probabilities

• NORM.DIST: Normal probabilities

• NORM.S.DIST: Standard normal probabilities

• EXPON.DIST: Exponential probabilities

• COMBIN: Number of combinations

• FACT: Factorial

Practice Problems and Solutions

• See assignment solutions for step-by-step calculations using the above formulas and Excel functions.

• Practice includes constructing sample spaces, calculating probabilities for various distributions, and interpreting results in business contexts.

Additional info: These notes synthesize and expand upon the provided lecture content, including definitions, formulas, and practical business applications, to serve as a comprehensive study guide for exam preparation in business statistics.