6.5 Probability & Two-Way Frequency Tables

6.5 Probability & Two-Way Frequency Tables

Probability questions on the Digital SAT test your ability to calculate simple, joint, compound, and conditional probabilities. The vast majority of these questions present data formatted inside a two-way frequency (contingency) table.

The single most critical skill is identifying the correct denominator by carefully reading how the problem constrains the sample space.

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1. Probability Axioms & Basic Formulas

The probability of an event $E$ within a finite sample space $S$ with equally likely outcomes is:

$$P(E) = \frac{n(E)}{n(S)} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}$$

Fundamental Properties

1. Range Bounds: $0 \le P(E) \le 1$.
   • $P(E) = 0 \implies$ Impossible event.
   • $P(E) = 1 \implies$ Certain event.
2. Complement Rule: The probability that $E$ does not occur is: $$P(\text{not } E) = P(E^c) = 1 - P(E)$$

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2. The Addition Rule (Union of Events)

When evaluating the probability that event $A$ OR event $B$ occurs, we use the General Addition Rule:

$$P(A \text{ or } B) = P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

+-----------------------------------------------------------------------------+
|                        THE ADDITION RULE & VENN DIAGRAM                     |
|                                                                             |
|              +-------------------+   +-------------------+                  |
|              |      Set A        |   |       Set B       |                  |
|              |            ( A ∩ B )                      |                  |
|              |            ( Overlap)                     |                  |
|              +-------------------+   +-------------------+                  |
|                                                                             |
|   P(A or B) = P(A) + P(B) - P(A and B)                                      |
|   We MUST subtract P(A and B) once to prevent double-counting the overlap!  |
|                                                                             |
|   [ MUTUALLY EXCLUSIVE (DISJOINT) EVENTS ]:                                 |
|   If A and B cannot occur simultaneously, P(A and B) = 0:                   |
|   --> P(A or B) = P(A) + P(B)                                               |
+-----------------------------------------------------------------------------+

Worked Example 1: AP Course Overlap

Problem: In a senior class of $300$ students, $140$ are enrolled in AP Chemistry, $180$ are enrolled in AP Calculus, and $70$ are enrolled in both courses. If a senior is selected at random, what is the probability that the student is enrolled in AP Chemistry, AP Calculus, or both?

Step-by-Step Solution:

1. Identify the individual probabilities: $$P(\text{Chem}) = \frac{140}{300}, \quad P(\text{Calc}) = \frac{180}{300}, \quad P(\text{Chem} \cap \text{Calc}) = \frac{70}{300}$$
2. Apply the addition rule: $$P(\text{Chem} \cup \text{Calc}) = \frac{140}{300} + \frac{180}{300} - \frac{70}{300} = \frac{140 + 180 - 70}{300} = \frac{250}{300} = \frac{5}{6}$$

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3. Two-Way Frequency Tables: Anatomy & Navigation

A two-way table categorizes individuals according to two distinct qualitative variables.

+-----------------------------------------------------------------------------+
|                       TWO-WAY CONTINGENCY TABLE ANATOMY                     |
|                                                                             |
|                     Variable Y: Outcome 1   Variable Y: Outcome 2   Total   |
|   Variable X: Cat 1       Cell (1,1)              Cell (1,2)        Row 1   |
|   Variable X: Cat 2       Cell (2,1)              Cell (2,2)        Row 2   |
|   Total                   Col 1 Total             Col 2 Total     GRAND TOT |
|                                                                             |
|   - Joint Frequencies: The interior 4 cells (e.g. Cat 1 AND Outcome 1)      |
|   - Marginal Frequencies: The row totals and column totals                  |
|   - Grand Total (N): Bottom-right cell (sum of all observations)            |
+-----------------------------------------------------------------------------+

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4. Conditional Probability: Restricting the Denominator

Conditional probability calculates the likelihood of an event occurring given that another condition has already been met.

$$P(A \mid B) = \frac{n(A \cap B)}{n(B)} = \frac{P(A \cap B)}{P(B)}$$

+-----------------------------------------------------------------------------+
|                   THE THREE SAT PROBABILITY QUESTION TYPES                  |
|                                                                             |
|   Sample Table: Exam Performance & Review Session Attendance                |
|                                                                             |
|   Category                    Passed Exam    Did Not Pass    Row Total      |
|   -------------------------   ------------   -------------   -----------    |
|   Attended Review Session         72               8              80        |
|   Did Not Attend Review           48              32              80        |
|   Column Total                   120              40             160        |
|                                                                             |
|   [ TYPE 1: OVERALL / JOINT PROBABILITY ]                                   |
|   "What is the probability that a randomly chosen student attended review   |
|    AND passed the exam?"                                                    |
|    --> Denominator = GRAND TOTAL (160)  --> 72 / 160 = 0.45                 |
|                                                                             |
|   [ TYPE 2: CONDITIONAL ON ROW ]                                            |
|   "Given that a student ATTENDED REVIEW, what is the prob they passed?"     |
|    --> Denominator = ROW TOTAL (80)     --> 72 / 80 = 0.90                  |
|                                                                             |
|   [ TYPE 3: CONDITIONAL ON COLUMN ]                                         |
|   "If a student who PASSED is selected at random, prob they attended?"      |
|    --> Denominator = COLUMN TOTAL (120) --> 72 / 120 = 0.60                 |
+-----------------------------------------------------------------------------+

Denominator Selection Guide for SAT Word Problems

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5. Independent vs. Dependent Probability

Two events $A$ and $B$ are independent if the occurrence of $B$ does not affect the probability of $A$:

$$P(A \mid B) = P(A) \iff P(A \cap B) = P(A) \cdot P(B)$$

Sequential Probability: Replacement Rules

• With Replacement (Independent): The sample space $n$ remains constant on successive draws.
• Without Replacement (Dependent): Both the favorable count and the total sample space decrease on successive draws.

Worked Example 2: Probability Without Replacement

Problem: A jar contains 6 red marbles, 4 blue marbles, and 5 green marbles ($15$ total). Two marbles are selected at random without replacement. What is the probability that both marbles are blue?

Step-by-Step Solution:

1. First Draw: $P(\text{Blue}_1) = \frac{4}{15}$.
2. Second Draw (given first was blue): $3$ blue marbles remain out of $14$ total: $P(\text{Blue}_2 \mid \text{Blue}_1) = \frac{3}{14}$.
3. Apply the general multiplication rule: $$P(\text{Both Blue}) = \frac{4}{15} \times \frac{3}{14} = \frac{12}{210} = \frac{2}{35}$$

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6. Desmos Playbook for Complex Probability Calculations

+-----------------------------------------------------------------------------+
|                       DESMOS PROBABILITY SHORTCUTS                          |
|                                                                             |
|   1. Exact Fraction Simplification:                                         |
|      Type: 72 / 120  --> Click the fraction icon to display 3/5.            |
|                                                                             |
|   2. Multi-Step Dependent Multiplication:                                   |
|      Type: (4/15) * (3/14)  --> Displays 2/35 directly.                     |
|                                                                             |
|   3. Overlapping Union Arithmetic:                                          |
|      Type: (140 + 180 - 70) / 300  --> Displays 5/6 directly.               |
+-----------------------------------------------------------------------------+