Probability Concepts, Contingency Tables, and Event Relations

Course Logistics and Administrative Information

• Exams are specific exclusively to this individual section; no other section takes the exam in the manner conducted here.

• Course materials and modules on Canvas for this section are currently blocked out and not yet available.

Identifying Event Types and Conditional Probability Clues

• Specific phrasing in English provides structural clues to identify when an event is conditional versus marginal or joint.

• Common phrases that indicate conditional probability include "given that…", "given…", or sentences structured like "If…".

• When a statement provides the known outcome of an event or explicitly states that a prior event has occurred, that event serves as the condition.

• Mathematically, this condition corresponds to Event $B$ in the conditional probability expression $P(A \mid B)$.

Union and Intersection Probability Formulas

• Key probability operators map directly to English conjunctions:

  • The union symbol ($\cup$) corresponds to the word "or".

  • The intersection symbol ($\cap$) corresponds to the word "and".

• Phrasing with "or both" represents the standard inclusive union and does not alter the fundamental union formula.

• The standard formula for the union of two events is: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

• Rearranging the formula allows solving for the intersection when given the individual probabilities and the union: $P(A \cap B) = P(A) + P(B) - P(A \cup B)$

• Sample problem values:

  • Percentage of people who like peanut butter: $75\%$

  • Percentage of people who like jelly: $93\%$

  • Percentage of people who like apple: $72\%$

• Using three given component values allows calculating the unknown intersection probability directly.

Marginal, Joint, and Conditional Probabilities in Contingency Tables

• Fundamental relationship connecting joint, conditional, and marginal probabilities: $\text{Intersection} = \text{Conditional} \times \text{Marginal}$ $P(A \cap B) = P(A \mid B) \times P(B)$

• Marginal Probability:

  • Refers strictly to the probability of a single event occurring without any conditioning, intersection, or union ($P(A)$).

  • Contingency tables display raw frequencies that convert directly into relative frequencies representing probabilities.

  • Every marginal probability relies on table row or column totals divided by the total sample size.

  • Example (Right-Handedness):

  • Total number of right-handed individuals in the total row: $87$

  • Total sample size: $635$

  • Formula application: $P(\text{Right Handed}) = \frac{87}{635}$

• Joint Probability:

  • Represented by the numbers located in the interior (middle) cells of a contingency table.

  • Corresponds to the intersection of two events ("and" statements).

  • Example (Right-Handed Male):

  • Number of individuals who are right-handed and male: $43$

  • Joint probability calculation uses the overall sample size: $P(\text{Right Handed} \cap \text{Male}) = \frac{43}{635}$

  • Both marginal and joint probabilities use the overall sample size ($635$) as the denominator.

• Conditional Probability:

  • Denoted as $P(\text{Left Handed} \mid \text{Female})$.

  • Restricts the denominator to the subset specified by the given condition rather than using the overall sample size.

• Additional frequency counts noted from data sets:

  • Sum total count: $140 + 635$

  • Aspirin group total count: $317$

Sequence and Probability Event Problems

• Sequence Problem 1: Defining an event where an individual gets the first two questions correct.

• Sequence Problem 2: Defining an event where the first question an individual gets correct is specifically the fourth question (indicating incorrect answers on questions 1, 2, and 3, followed by a correct answer on question 4).

• Reference item number: Eight six ($8.6$).

Statistical Deviations and Event Relationships

• Variance and Standard Deviation Calculations:

  • The initial step requires calculating deviations from the mean.

  • The subsequent column in the calculation table requires squaring all of those deviation numbers.

• Disjoint vs. Independent Events:

  • Question: If two events are disjoint, are they independent?

  • Answer: No.

  • Disjoint (mutually exclusive) events cannot be independent. If two events are disjoint ($A \cap B = \emptyset$), knowing that one event has occurred guarantees that the other event cannot occur, establishing a dependence relationship.

• Specific numeric threshold: $21$ (noting that one does not want to be $21$).