10.1 Theoretical vs. Experimental Probability, Sample Spaces, & Single-Event Outcomes

10.1 Theoretical vs. Experimental Probability, Sample Spaces, & Single-Event Outcomes

Probability is the branch of mathematics that quantifies uncertainty, measuring the chance or likelihood that a specific event will happen. On the TABE 13&14 Mathematics assessment (Levels M, D, and A), probability problems test your ability to model single-stage experiments, calculate exact theoretical ratios, evaluate empirical data from experimental trials, construct sample spaces, and apply the Fundamental Counting Principle.

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The Likelihood Scale & Numerical Representation

The probability of any event $E$, denoted as $P(E)$, is a real number bounded strictly between $0$ and $1$, inclusive:

$0 \le P(E) \le 1$

Probabilities can be represented interchangeably in three mathematical formats: simplified fractions, decimals, and percentages.

[!IMPORTANT] The Sum of Elementary Probabilities: In any valid probability experiment, the sum of the probabilities of all distinct, non-overlapping simple outcomes in the sample space must equal exactly $1$ (or $100%$): $\sum P(e_i) = P(e_1) + P(e_2) + \dots + P(e_n) = 1$

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Theoretical Probability of Single Events

Theoretical probability is calculated mathematically under the assumption that all possible outcomes in an experiment are equally likely (fair). It represents the expected likelihood before any physical trials occur.

The Theoretical Probability Formula

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

Standard Probability Models on TABE

1. Standard 6-Sided Die: Outcomes $S = {1, 2, 3, 4, 5, 6}$, where each number has $P = \frac{1}{6}$.
2. Standard 52-Card Deck: $52$ total cards, split into $2$ colors (Red $26$, Black $26$), $4$ suits ($13$ Hearts, $13$ Diamonds, $13$ Spades, $13$ Clubs), and $3$ face cards per suit (Jack, Queen, King $\implies 12$ total face cards).
3. Fair Spinners & Urns: Circular wheels partitioned into congruent sectors or containers with colored marbles/tokens.

Worked Example 1: Standard Card Deck Selection

Problem: A single card is drawn at random from a standard, shuffled 52-card deck. What is the theoretical probability of drawing a red face card?

1. Identify the Sample Space Size: $n(S) = 52$.
2. Count Favorable Outcomes: Face cards are Jacks, Queens, and Kings. There are $3$ red Hearts face cards and $3$ red Diamonds face cards $\implies n(\text{Red Face}) = 3 + 3 = 6$.
3. Apply the Formula & Simplify: $P(\text{Red Face}) = \frac{6}{52} = \frac{3}{26} \approx 0.1154 = \mathbf{11.54\%}$

Worked Example 2: Prime Numbers on a Die

Problem: A standard fair six-sided die is rolled once. What is the probability of rolling a prime number?

1. Sample Space: $S = {1, 2, 3, 4, 5, 6} \implies n(S) = 6$.
2. Identify Prime Outcomes: Prime numbers have exactly two distinct positive divisors ($1$ and itself). Primes in $S$ are ${2, 3, 5}$ (note that $1$ is neither prime nor composite) $\implies n(\text{Prime}) = 3$.
3. Calculate: $P(\text{Prime}) = \frac{3}{6} = \frac{1}{2} = \mathbf{0.50 = 50\%}$

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Experimental Probability & The Law of Large Numbers

Experimental probability (empirical probability) is calculated from the observed results of an actual experiment or historical data collection.

The Experimental Probability Formula

$P(\text{exp}) = \frac{\text{Number of Times the Event Occurred}}{\text{Total Number of Trials Conducted}}$

The Law of Large Numbers (LLN)

The Law of Large Numbers states that as the number of repetitions or trials in a probability experiment increases ($n \to \infty$), the relative frequency (experimental probability) gets closer and closer to the theoretical probability.

Law of Large Numbers (Coin Toss Relative Frequency Convergence):

Relative Frequency of Heads
 1.0 | *
 0.8 |   * 
 0.6 |     *   *   * ---------------------------- Theoretical P(Heads) = 0.50
 0.4 |           *   * * * * * * * * * * * * * *
 0.2 | 
 0.0 +-------------------------------------------
     0   10  50  100 250 500 1,000 5,000 10,000   (Number of Trials n)

Worked Example 3: Quality Control Defect Prediction

Problem: A manufacturing plant tests $400$ computer motherboards and discovers $18$ have defective solder joints. If the factory produces $14,000$ motherboards this month, what is the best estimate of the total number of defective motherboards produced?

1. Calculate the Experimental Probability: $P(\text{Defective}) = \frac{18}{400} = \frac{9}{200} = 0.045 = 4.5\%$
2. Multiply by the Total Batch Volume: $\text{Estimated Defects} = P(\text{Defective}) \times \text{Total Batch} = 0.045 \times 14,000 = \mathbf{630\text{ motherboards}}$

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Sample Spaces & Systematic Counting Models

A sample space ($S$) is the comprehensive set containing all possible outcomes of a probability experiment.

Methods for Representing Sample Spaces

1. Roster Notation (Listing): Explicitly listing all elements inside set brackets: $S = {H, T}$.
2. Two-Dimensional Outcome Grids: A grid mapping the outcomes of two simultaneous stages.
3. Tree Diagrams: A branching visualization showing sequential stages and path probabilities.

Two-Stage Tree Diagram (Tossing a Coin then Rolling a 4-Sided Die):

Stage 1 (Coin)        Stage 2 (Die)        Outcome Path
                      ├── 1 -------------> (H, 1)
                      ├── 2 -------------> (H, 2)
        ├── Heads ────┼── 3 -------------> (H, 3)
        │             └── 4 -------------> (H, 4)
Start ──┤
        │             ├── 1 -------------> (T, 1)
        │             ├── 2 -------------> (T, 2)
        └── Tails ────┼── 3 -------------> (T, 3)
                      └── 4 -------------> (T, 4)
Total Outcomes in Sample Space = 2 × 4 = 8

Outcome Grid for Rolling Two Standard Six-Sided Dice ($36$ Outcomes)

When rolling two dice (Die 1 along rows, Die 2 along columns), the sum of the two faces produces the classic triangular distribution:

| Die 1 \ Die 2 | 1 | 2 | 3 | 4 | 5 | 6 | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | 1 | $(1,1)=2$ | $(1,2)=3$ | $(1,3)=4$ | $(1,4)=5$ | $(1,5)=6$ | $(1,6)=7$ | | 2 | $(2,1)=3$ | $(2,2)=4$ | $(2,3)=5$ | $(2,4)=6$ | $(2,5)=7$ | $(2,6)=8$ | | 3 | $(3,1)=4$ | $(3,2)=5$ | $(3,3)=6$ | $(3,4)=7$ | $(3,5)=8$ | $(3,6)=9$ | | 4 | $(4,1)=5$ | $(4,2)=6$ | $(4,3)=7$ | $(4,4)=8$ | $(4,5)=9$ | $(4,6)=10$ | | 5 | $(5,1)=6$ | $(5,2)=7$ | $(5,3)=8$ | $(5,4)=9$ | $(5,5)=10$ | $(5,6)=11$ | | 6 | $(6,1)=7$ | $(6,2)=8$ | $(6,3)=9$ | $(6,4)=10$ | $(6,5)=11$ | $(6,6)=12$ |

TABE Question Application: What is the probability of rolling a sum of $7$ with two dice? There are $6$ favorable diagonal pairs: $(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)$. Thus, $P(\text{Sum of } 7) = \frac{6}{36} = \frac{1}{6}$.

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The Fundamental Counting Principle

The Fundamental Counting Principle (Multiplication Counting Principle) states that if a compound process can be broken down into $k$ consecutive independent stages, where Stage 1 has $n_1$ possible outcomes, Stage 2 has $n_2$ possible outcomes, and Stage $k$ has $n_k$ possible outcomes, the total number of distinct combined outcomes is the product of the individual stage counts:

$\mathbf{\text{Total Outcomes} = n_1 \times n_2 \times n_3 \times \dots \times n_k}$

Worked Example 4: Security Passcode Creation

Problem: A warehouse security gate requires a 4-character employee access code. The first character must be a letter from $A$ through $E$ (5 options). The remaining three characters are single-digit numbers ($0$ through $9$). How many unique passcodes can be created if (a) digits may repeat, and (b) digits cannot repeat?

• Case A: Repetition Allowed $\text{Total Codes} = 5 \times 10 \times 10 \times 10 = \mathbf{5,000\text{ codes}}$
• Case B: No Digits Repeated $\text{Total Codes} = 5 \times 10 \times 9 \times 8 = \mathbf{3,600\text{ codes}}$

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Complementary Events & The "At Least One" Rule

The complement of an event $A$, denoted $A'$ (or $A^c$ or $\text{not } A$), consists of all outcomes in the sample space $S$ that are not contained in event $A$.

$\mathbf{P(A) + P(A') = 1} \qquad \Longleftrightarrow \qquad \mathbf{P(A') = 1 - P(A)}$

Worked Example 5: Probability of Complementary Events

Problem: A game spinner is divided into $10$ equal sectors: $4$ red, $3$ blue, $2$ yellow, and $1$ green. What is the probability of spinning the arrow and landing on a color that is not red?

1. Find $P(\text{Red}): P(\text{Red}) = \frac{4}{10} = 0.40$.
2. Apply the complement formula: $P(\text{Not Red}) = 1 - P(\text{Red}) = 1 - \frac{4}{10} = \frac{6}{10} = \frac{3}{5} = \mathbf{0.60 = 60\%}$

The "At Least One" Rule

When solving complex multi-stage problems where calculating direct probabilities requires summing many branches, the complement provides an efficient shortcut:

$\mathbf{P(\text{At Least One Success}) = 1 - P(\text{Zero Successes})}$