13.3 Theoretical and Experimental Probability & Compound Events
Key Takeaways
- Probability quantifies likelihood on a closed scale from 0 (impossible event) to 1 (certain event), with theoretical probability defined as P(A) = (number of favorable outcomes) / (total equally likely outcomes).
- Experimental (empirical) probability is the observed ratio of successes to trials; by the Law of Large Numbers, experimental probability converges toward theoretical probability as the number of trials approaches infinity.
- The Complement Rule dictates that P(A') = 1 - P(A); it is especially powerful for solving 'at least one' compound problems via P(at least one) = 1 - P(none).
- The Fundamental Counting Principle states that if k successive events have n_1, n_2, ..., n_k independent choices, the total compound outcomes equal n_1 × n_2 × ... × n_k.
- Compound probability rules distinguish between independent events (P(A and B) = P(A) · P(B)), dependent events (P(A and B) = P(A) · P(B|A)), mutually exclusive events (P(A or B) = P(A) + P(B)), and overlapping events (P(A or B) = P(A) + P(B) - P(A and B)).
Theoretical and Experimental Probability & Compound Events
Quick Answer: On the WEST-B Mathematics subtest (Objective 0016), probability questions require applying exact rules of counting and chance. Core formulas include: Theoretical Probability ($P(A) = \frac{\text{favorable}}{\text{total}}$), Complement Rule ($P(A') = 1 - P(A)$), Fundamental Counting Principle ($n_1 \times n_2 \times \dots \times n_k$), Independent Multiplication ($P(A \text{ and } B) = P(A) \times P(B)$), Dependent Multiplication ($P(A \text{ and } B) = P(A) \times P(B|A)$), Overlapping Addition Rule ($P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$), and the 'At Least One' Rule ($P(\ge 1) = 1 - P(\text{none})$).
1. Probability Foundations & Definitions
Probability is the mathematical measure of the likelihood that an event will occur.
+---------------------------------------------------------------------------------------------------+
| PROBABILITY SCALE & TERMINOLOGY |
| |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | TERM | DEFINITION | MATHEMATICAL NOTATION / EXAMPLE | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Experiment | A repeatable process that | Tossing a coin, rolling a die, | |
| | | yields an outcome | drawing a card | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Sample Space (S) | The set of ALL possible | Rolling a 6-sided die: | |
| | | distinct outcomes | S = {1, 2, 3, 4, 5, 6}, |S| = 6 | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Event (E) | Any subset of sample space S| Rolling an even number: | |
| | | | E = {2, 4, 6}, |E| = 3 | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Probability Axiom | Probability must fall | 0 ≤ P(E) ≤ 1 | |
| | | strictly between 0 and 1 | (or 0% to 100%) | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Impossible Event | An event that cannot occur | P(∅) = 0 (e.g., rolling a 7 on die) | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Certain Event | An event guaranteed to occur| P(S) = 1 (e.g., rolling ≤ 6 on die) | |
| +-----------------------+-----------------------------+-------------------------------------+ |
+---------------------------------------------------------------------------------------------------+
THE PROBABILITY SCALE
Impossible Unlikely Even Chance Likely Certain
|----------------------|---------------------|-------------------|--------------------|
0.0 0.25 0.50 0.75 1.0
0% 25% 50% 75% 100%
0 1/4 1/2 3/4 1
2. Theoretical vs. Experimental Probability & Law of Large Numbers
+---------------------------------------------------------------------------------------------------+
| THEORETICAL VS. EXPERIMENTAL PROBABILITY MATRIX |
| |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | PROPERTY | THEORETICAL PROBABILITY | EXPERIMENTAL (EMPIRICAL) PROBABILITY|
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Calculation Basis | Mathematical analysis of | Actual collected data from trials/ | |
| | | equally likely outcomes | experiments | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Formula | P(A) = n(A) / n(S) | P_exp(A) = (Observed Successes) / | |
| | | | (Total Number of Trials) | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Example: Fair Coin | P(Heads) = 1/2 = 0.50 | 54 Heads in 100 flips = 54/100=0.54 |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Law of Large Numbers | Fixed theoretical benchmark | As trials n → ∞, Experimental P | |
| | | value | approaches Theoretical P | |
| +-----------------------+-----------------------------+-------------------------------------+ |
+---------------------------------------------------------------------------------------------------+
The Complement Rule
The complement of an event $A$ (denoted $A'$, $A^c$, or $\text{not } A$) consists of all outcomes in sample space $S$ that are not in $A$:
- The 'At Least One' Shortcut: When calculating the probability that an event occurs at least once across multiple trials:
3. Counting Principles, Permutations & Combinations
A. The Fundamental Counting Principle (Multiplication Rule)
If an experiment consists of $k$ consecutive sequential steps, where step 1 can occur in $n_1$ ways, step 2 in $n_2$ ways, ..., and step $k$ in $n_k$ ways, the total number of compound outcomes in the sample space is: Example: A school cafeteria offers a lunch combo with a choice of 3 sandwiches, 4 sides, and 2 drinks. Total distinct lunches $= 3 \times 4 \times 2 = 24$.
B. Permutations vs. Combinations
+---------------------------------------------------------------------------------------------------+
| PERMUTATION VS. COMBINATION MATRIX |
| |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | FEATURE | PERMUTATIONS P(n, r) | COMBINATIONS C(n, r) |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Does Order Matter? | YES (Order / Position counts| NO (Order does not matter; group/ |
| | | e.g., President, VP, Sec) | committee selection) |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Mathematical Formula | P(n, r) = n! / (n - r)! | C(n, r) = n! / [r! (n - r)!] |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Example: Select 2 | P(5, 2) = 5 × 4 = 20 | C(5, 2) = (5 × 4) / (2 × 1) = 10 |
| | items from 5 {A,B,...}| (AB is distinct from BA) | ({A, B} is same group as {B, A}) |
| +-----------------------+-----------------------------+-------------------------------------+ |
+---------------------------------------------------------------------------------------------------+
4. Compound Events: Independent vs. Dependent
A compound event involves the simultaneous or sequential occurrence of two or more simple events.
+---------------------------------------------------------------------------------------------------+
| COMPOUND PROBABILITY RULES |
| |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | EVENT RELATIONSHIP | CONDITION | PROBABILITY FORMULA |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Independent Events | Outcome of Event A does NOT | P(A and B) = P(A) × P(B) |
| | (With Replacement) | affect Event B | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Dependent Events | Outcome of Event A ALTERS | P(A and B) = P(A) × P(B | A) |
| | (Without Replacement) | the sample space of Event B | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Mutually Exclusive | Events CANNOT occur at the | P(A or B) = P(A) + P(B) |
| | (Disjoint) Events | same time: P(A and B) = 0 | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Overlapping Events | Events CAN occur | P(A or B) = P(A) + P(B) - P(A and B)|
| | (Non-Disjoint) | simultaneously | |
| +-----------------------+-----------------------------+-------------------------------------+ |
+---------------------------------------------------------------------------------------------------+
Venn Diagram: Overlapping vs. Disjoint
Overlapping Events (A ∩ B ≠ ∅) Mutually Exclusive (Disjoint)
+----------+ +----------+ +----------+ +----------+
/ Only A \ / Only B \ / \ / \
| X | | Event A | Event B |
\ (A) / \ (B) / \ / \ /
+----------+ +----------+ +----------+ +----------+
Intersection
(A and B)
5. Tree Diagrams & Geometric Probability
A. Tree Diagrams
A tree diagram visualizes multi-stage probabilistic experiments by displaying branches for each sequential outcome with conditional probabilities labeled on each branch.
Tree Diagram: Two Coin Flips
Outcome Probability
+-------+ +-----------+
H -->| H, H | 1/2 × 1/2 = 1/4
1/2 / +-------+ +-----------+
+----> H
/ \ 1/2 +-------+ +-----------+
/ T -->| H, T | 1/2 × 1/2 = 1/4
Start +-------+ +-----------+
\ 1/2 +-------+ +-----------+
\ 1/2 H -->| T, H | 1/2 × 1/2 = 1/4
+----> T +-------+ +-----------+
\ 1/2 +-------+ +-----------+
T -->| T, T | 1/2 × 1/2 = 1/4
+-------+ +-----------+
B. Geometric (Area) Probability
In geometric probability, outcomes correspond to points in a geometric region (length, area, or volume):
- Concentric Circles Example: If a bullseye has radius $r$ inside a dartboard of radius $R$:
6. Step-by-Step Worked Probability Problems
Problem 1: Dependent Probability (Without Replacement)
Problem: A jar contains $6$ red marbles, $4$ blue marbles, and $2$ yellow marbles ($12$ marbles total). Two marbles are drawn at random one after another without replacement. What is the probability that both marbles drawn are red?
Step-by-Step Solution:
- Calculate probability of drawing first red marble:
- Calculate conditional probability of drawing second red marble:
- One red marble has been removed: $5$ red marbles remaining.
- Total marbles remaining: $12 - 1 = 11$ marbles.
- Multiply probabilities (Multiplication Rule for Dependent Events):
- Convert to decimal/percentage:
Problem 2: Overlapping Addition Rule (Deck of Cards)
Problem: A single card is drawn at random from a standard, well-shuffled 52-card deck. What is the probability that the card drawn is either an Ace OR a Heart?
Step-by-Step Solution:
- Identify individual event probabilities:
- Event $A$ (Drawing an Ace): There are $4$ Aces in a deck $\implies P(A) = \frac{4}{52}$.
- Event $B$ (Drawing a Heart): There are $13$ Hearts in a deck $\implies P(B) = \frac{13}{52}$.
- Identify overlapping outcomes ($A \cap B$):
- There is exactly $1$ card that is both an Ace and a Heart (the Ace of Hearts).
- Apply the Addition Rule for Overlapping Events:
- Simplify the fraction:
Problem 3: The 'At Least One' Complement Problem
Problem: A couple plans to have $4$ children. Assuming each child is equally likely to be a boy or a girl ($P(\text{Boy}) = P(\text{Girl}) = 0.5$) independently, what is the probability that the couple has at least one girl?
Step-by-Step Solution:
- Define the complementary event:
- The complement of "at least one girl" is "zero girls" (which means all 4 children are boys).
- Calculate probability of zero girls (all 4 boys):
- Apply Complement Rule:
- Convert to decimal/percentage: There is a $\frac{15}{16}$ ($93.75%$) chance of having at least one girl.
A bag contains 5 red marbles, 4 blue marbles, and 3 green marbles (12 marbles total). If two marbles are drawn at random one after another without replacement, what is the probability that both marbles drawn are blue?
A standard 52-card deck is shuffled, and a single card is dealt at random. What is the probability that the card drawn is either a Face card (Jack, Queen, King) OR a Diamond?
A fair six-sided die is rolled 3 times. What is the probability of rolling at least one 6 across the 3 rolls?
A high school club must select a 4-person executive leadership team consisting of 1 president, 1 vice president, 1 secretary, and 1 treasurer from a pool of 8 candidates. If no student can hold more than one office, how many different leadership slates are possible?