Business Statistics: Probability & Distributions

• Business Statistics Unit 4 covers probability concepts and distributions for the second semester of AY 2024-2025.
• The unit explores probability concepts, discrete probability distributions, and continuous probability distributions.

Probability Concepts

• Random experiments, probability, rules of probability, independent events, contingency tables, tree diagrams, Bayes' Theorem, and counting rules will be covered.

Discrete Probability Distributions

• Discusses discrete distributions, uniform distribution, binomial distribution, Poisson distribution, and hypergeometric distribution.

Continuous Probability Distributions

• Covers continuous distributions, uniform continuous distribution, normal distribution, and exponential distribution.

Random Experiments

• A random experiment describes an observational process, where the results are unknown in advance.
• The set of all possible outcomes is referred to as the sample space, denoted by S.
• Sample spaces with a countable number of outcomes are discrete.
• Discrete sample spaces can be enumerated easily or described by a rule if they are immense.

Examples of Sample Spaces

• Customer payment methods: S = {cash, debit card, credit card, check, e-wallet}
• Loan application outcomes: S = {approved, not approved}
• Rolling a die: S = {1, 2, 3, 4, 5, 6}
• Rolling two dice: S = {1-1, 1-2, 1-3, ..., 6-5, 6-6}, with 36 outcomes

Discrete and Continuous Measurements

• Some discrete measurements are best described by a rule, such as the number of hits on Google for a topic: S = {X = 0, 1, 2, 3, ...}.
• If the experiment yields a continuous measurement, its sample space cannot be listed but can be described by a rule, for example, a student's GPA: S = {all X such that 0.0 <= X <= 4.0}.
• An event is any subset of outcomes in the sample space.
• A simple, or elementary, event is a single outcome.
• A discrete sample space S consists of all simple events denoted by E₁, E₂, ..., Eₙ; thus, S = {E₁, E₂, ..., Eₙ}.
• Tossing a balanced coin has a sample space S = {heads, tails}.
• Observing a head or tail on a balanced coin is equally likely

Sample Space of Lottery Tickets

• Buying a lottery ticket has a sample space S = {win, lose} consisting of 2 elementary events.
• Winning and losing are not equally likely in a lottery.
• Simple events form building blocks for compound events.
• A compound event is composed of two or more simple events.
• On Amazon's "Books & Music" website, the sample space for shopper choices has seven categories: S = {Books, DVD, VHS, Magazines, Newspapers, Music, Textbooks}.
• "Electronic media" (A = {Music, DVD, VHS}) and "Print periodicals" (B = {Newspapers, Magazines}) represent compound events within this sample space.
• Venn diagrams can visually show compound events.

Probability

• The probability of an event measures the likelihood of its occurrence.
• The probability of event A, denoted P(A), falls within the interval from 0 to 1, or 0 <= P(A) <= 1.
• P(A) = 0 indicates the event cannot occur.
• P(A) = 1 means the event is certain to occur.
• In a discrete sample space, the probabilities of all simple events must sum to 1: P(S) = P(E₁) + P(E₂) + ... + P(Eₙ) = 1.

Example Calculation of Probabilities

• Purchases are 32% credit card, 15% debit card, 25% cash, 8% check, and 20% e-wallet: P(S) = P(CR) + P(DR) + P(CU) + P(CA) + P(EW) = 0.32 + 0.15 + 0.25 + 0.08 + 0.20 = 1.00 There are 3 distinct ways of assigning probability:
• Empirical approach
• Classical approach
• Subjective approach

Empirical Approach

• Empirical data are collected through observation or experiments.
• Probabilities are based on the relative frequency of observed outcomes.
• Empirical estimation is necessary when there is no prior knowledge of the events.
• Estimates become more accurate with an increasing number of observations or experiments, as described by the law of large numbers.
• In flipping a coin 50 times, the proportion of heads should approach 0.50.
• When rolling two dice, the likelihood of getting a 7 is 6/36, irrespective of previous rolls.
• The law of large numbers suggests that as trials increase, the empirical probability approaches its theoretical limit.

Classical Approach

• A priori probabilities rely on logic, reason, or theory, not on experiments or experiences.
• The process allows envisioning the entire sample space as equally likely outcomes.
• A priori refers to assigning probabilities before observing the event.
• A priori calculations are rare in business situations, instead applying to cards, lottery numbers, and roulette, or in physical situations, assume the probability is proportional to the ratio of area, volume, or length.

Subjective Approach

• Subjective probability reflects personal judgment or expert opinion about an event's likelihood.
• Such judgment typically comes from experience with similar events and knowledge of causal processes.
• Subjective probability shares commonality with empirical probabilities, but has an informal and unquantified empirical basis.

Rules of Probability: Complement of an Event

• An event's complement, denoted A', includes everything in the sample space S except event A.
• Since A and A' together comprise the sample space, P(A) + P(A') = 1.
• To find P(A'), subtract P(A) from 1: P(A') = 1 – P(A).
• It is reported that 33% of all new businesses fail within the first 2 years; thus, the probability a new small business survives at least the first 2 years is: P(survive) = 1 – P(fail) = 1 – 0.33 = 0.67 or 67%.

Union of Two Events

• The union of two events includes all outcomes in the sample space S contained in event A, event B, or both.
• Denoted A ∪ B or "A or B", the symbol ∪ means either or both events occur.
• When a card is randomly chosen, Q ∪ R consists of getting a queen (4/52) or a red card (26/52) or both (2/52), where Q is the event that we draw a queen and R is the event that we draw a red card.

Intersection of Two Events

• The intersection of events A and B includes sample space outcomes found in both A and B.
• Denoted A ∩ B or "A and B".
• The probability of A ∩ B is called the joint probability and is denoted P(A ∩ B).
• ∩ means "and" since it means both events occur.
• If Q is drawing a queen and R is drawing a red card, then Q ∩ R is getting a card that is both a queen and red, which is two cards consisting of the Queen of Hearts and Queen of Diamonds.

The General Law of Addition

• The probability of the union of two events A and B is the sum of probabilities less the probabilities of their intersection: P(A U B) = P(A) + P(B) - P(A ∩ B)
• The intersection is counted twice when probabilities are added, leading to deduction that avoid overstating the probability of the union of events.
• P(Q U R) = P(Q) + P(R) - P(Q ∩ R) = (4/52) + (26/52) – (2/52) = 28/52 = 0.5385 or 53.85%

Mutually Exclusive Events

• Events A and B are mutually exclusive or disjoint if their intersection is the null set (no elements).
• Mutually exclusive means one event precludes the other.
• The null set is denoted by Ф, then P(A ∩ B) = 0.
• Mutually exclusive events do not exist in both categories and don't overlap, like age categories of being below 21 or being over 65.

Special Law of Addition

• If A and B are mutually exclusive events, then P(A ∩ B) = 0, and the addition reduces to P(A U B) = P(A) + P(B).
• It is given that P(under 21) = 0.28 and P(over 65) = 0.12; then P(under 21 or over 65) = 0.28 + 0.12 = 0.4 or 40%.

Collectively Exhaustive Events

• Events are deemed collectively exhaustive when their union comprises the entire sample space.
• Binary or dichotomous events are those that are mutually exclusive and collectively exhaustive, such as whether a car repair is covered by warranty(A) or not (A').
• There can be more than two mutually exclusive, collectively exhaustive events such as credit card, debit card, cash, check and e-wallet as customer payment methods.
• Categorical data can be collapsed into binary events by defining the second category as everything not in the first category
• It is useful to assign a numerical value (0 or 1) to each binary event, P(A ∩ B).

Conditional Probability

• A conditional probability is defined as event A, given that event B occurred, denoted P(A|B), read as "the probability of A given B".
• The vertical line is read as "given".
• Conditional probability equals the joint probability of A and B divided by the probability of B: P(A|B) = P(A ∩ B) / P(B) for P(B) > 0.
• The logic restricts sample space, S, to B (an event that occurred). The intersection, A and B, is the part of B also in A. The ratio of sizes between set A and B to set B refers to conditional probability of P(A | B).

Conditional Probability Example

• In a population aged 16-21 and not in college, 13.50% are unemployed, 29.05% are high school dropouts, and 5.32% are unemployed high school dropouts.
• The conditional probability of being unemployed, given the person is a high school dropout is:
  • U = the event that the person is unemployed
  • D = the event that the person is a high school dropout
  • P(U) = 0.1350
  • P(D) = .2905
  • P(U ∩ D) = 0.0532
  • P(U | D) = P(U ∩ D) / P(D) = 0.0532 / 0.2905 = 0.1831 or 18.31%
• The likelihood of being unemployed is greater for high school dropouts. Being a dropout changes one's probability of unemployment.
• Events depend on P(A) differing from P(A | B).

Dependent Events

• While dependent events may prove causally related, statistical dependence doesn't show cause and effect, only that event B affecting event A.
• Based on data, dessert (D) is ordered with a 0.08 probability, bottled beverage (B) is ordered with 0.14 probability.
• The joint probability suggests a 0.0112 probability of ordering both.
• P(D) = 0.08, P(B) = 0.14, and P(D ∩ B) = 0.0112; the equation P(D | B) = P(D ∩ B) / P(B) equals 0.0112 / 0.14 = 0.08% of 0.08.
• P(D | B) = 0.08 leads to the conclusion that D & B are independent.
• P(D)P(B) = 0.08*0.14 = 0.0112 being equal to P(A ∩ B) = 0.0112 verifies independent events. If the customer ordered bottled beverage that doesn't change the probability that he or she will also order the dessert.

Independent Events

• The probability that several independent events occur simultaneously is the multiplication law for independent events, applicable to system reliability, given by P(A₁ A₂ ... Aₙ) = P(A₁) P(A₂)...P(Aₙ).
• If two independent file servers each have 99% reliability, or 0.99 probability, two servers dramatically increase the system availability to 99.99%, in which case, dual servers dramatically improve reliability to 99.99%.
• Let F₁ denote the event that server 1 fails, and F₂ denote the event that server 2 fails, then P(F₁) = 1 – 0.99 = 0.01 and P(F₂) = 1 – 0.99 = 0.01, making P(F₁ F₂) = P(F₁) P(F₂) = 0.01 * 0.01 = 0.0001
• The probability that at least one server is up is 1 minus the probability that both servers are down, i.e., 1 – 0.0001 = 0.9999

Redundancy

• With low component availability high availability can be achieved.

NASA space shuttle

• The NASA space shuttle consists of three flight computers, the chance of failure is 0.03 of 100 missions.

Five Nines Rule

• The Fives Rule implies only about 5 minutes downtime for repairs per year

Contingency Tables

• A contingency table is a cross-tabulation of frequencies in rows and columns; the cell at each intersection then reveals a frequency.
• The contingency table has two variables in rows and columns and is applied to a survey.
• An example used a table showing tuition costs and 5-year salary gains.
• Salary gains include salary minus the sum of tuition with foregone compensation.
• 67 schools were examined: graduates of high schools have higher salaries than graduates of lower paid schools.
• The table reveals that MBA graduates tend to have that extra edge: high 15/67, lower tuition fees 32 of 67.
• The marginal probabilities of an event are found by the rows or sample size.
• If 33 out of 67 schools had a medium salary gain, then the marginal probability of P(S₂) is 33/67=0.4925, in other words gains for schools with $50k and 100k were 49%
• probability of low tuition P(T₁)= 16/66=.2388 and 24% of top tier students has that tuition fee is capped at under $40K.
• All nine cells are also to calculate the joint probability which can be written ad the intersecting events like so P(T₁S3)=0.0149- at most, there is a top tier school with lesser fees.
• The conditionals are also restricted in a single row with a condition
• The approach is used for joint probability by using 9 cells for relative frequency to be conditional

Tree Diagrams

• Business planning activity includes events and probabilities of a tree diagram.
• Marginal probabilities of funds are a good start as they help to visualize all possible outcomes.
• To do joint probability, all probabilities in a terminal event is multiplied along the branch.

Bayes' Theorem

• Thomas Bayes wrote an important theorem in 1761 which then provided methods of using probabilities.
• An unconditioned value has event B is revised after the A is considered
• The unconditional is after its conditional probability.

Counting Rules

• Event A occurs in n1 way and event B occurs and then even A&B both happen = N1 and N2
• There are m events that occur with N1, N2... Nm

Example counting stock labels

• Unique unique stock-keeping labels are using 2 letters which range from AA ro ZZ, the 4 number and digit ranges can be from 0 to 9. Box 1= 26 letters A to Z
• Box 2= 26 letters A to Z
• Box 3 =, 10 digits 0 to 9
• Box 4=, 10 digits 0 to 9
• Box 5 =. 10 digits 0 to 9
• Box 6= 10 digits 0 to 9
• All the stock keeping labels adds to 6760,00 with 2626101010*10
• Factorials are unique if everything is multiplied by 1 instead
• Ex: Home appliance service trucks make 3 home stops - how can this be arranged?
• 3!= 3x2x1 =6
• Permutations are another way if something must be arranges
• Ex= Five home appliance trucks need calls but only 3 trucks will call
• nPr =5! / (5-3)!= (54321) / 2*1= 120/2=60
• There a 10 unique groups with 3 customers in groups of 3
• ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE
• ABC ACB BCA BAC CAB
• Combination is the the collection or items are is chosen at random and the replacements don't matter
• NCr = n! / r!(n-r)!= 5! / 3!(5-3)! = 54321 / (321)(2*1)= 120/12=10
• combination is fewer then permutations (this is because order is not a consideration)

Discrete vs Continuous

• Discrete the difference involves the computing of probabilities.
• the probability function can use the variant values and the probability of the curve can be made of graphics.

Discrete Probability

• discrete uniformity

Bivariate

• know 2 variables and find the probabilities.

Binomial

• Bernouilli Process
• Fixed Process
• All Process - (if those steps above occur)

Possion

• The Number of successes happens and then that interval is a space of it's own with equal sizes.

Hypergeometric

• The random number and trial probability can be on each group
• There can be more groups prefer one choice to another

Uniform formula application

• Uniform is a prob density with f(X)
• There is total area
• ex) the 2000/5000 formula is uniform with total gallons, what will station sell given that it is >= 4000?

Normal

• This distribution can be found on its means and two parameters can always be found, find the probability between calls.
• (6.3+ 2.2)

Exponential

• Lambda can be known and its distribution is its means for its means deviation.
• time hours are found for hours and breakdowns (of machines)