9.1 Simple Probability & Theoretical vs. Experimental

9.1 Simple Probability & Theoretical vs. Experimental

Quick Summary: Simple probability quantifies the likelihood of a single event occurring. Theoretical probability $P(E) = \frac{n(E)}{n(S)}$ determines exact likelihood by analyzing sample spaces under the assumption of equiprobable outcomes. Experimental probability $P_{\text{exp}}(E) = \frac{f}{N}$ calculates relative frequency from empirical data gathered across $N$ trials. The Law of Large Numbers bridges these concepts: as the trial count $N$ increases toward infinity, experimental results converge to the theoretical expectation. All probabilities are bounded between $0$ (impossible) and $1$ (certain), and complementary events satisfy $P(\text{not } E) = 1 - P(E)$.

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The Probability Continuum & Core Terminology

Probability is the mathematical branch that measures uncertainty. In the Ontario Mathematics Curriculum (Strand D: Data), probability concepts begin with qualitative descriptions (impossible, unlikely, equally likely, likely, certain) and progress to precise numerical values.

The Mathematical Scale of Probability

Any probability value $P(E)$ for an event $E$ must satisfy the fundamental inequality:

$0 \le P(E) \le 1 \quad \text{or equivalently} \quad 0\% \le P(E) \le 100\%$

• Impossible Event ($P = 0$ or $0%$): An outcome that cannot occur under any circumstances (e.g., rolling a $7$ on a standard fair six-sided die).
• Certain Event ($P = 1$ or $100%$): An outcome that must occur on every single trial (e.g., drawing a red or black card from a standard deck).
• Equally Likely Events ($P = 0.5$ or $50%$): An outcome with a 50-50 chance of occurrence (e.g., flipping a fair coin and getting Heads).

MPT Test Tip: On the Ontario MPT, probability answers may appear as simplified fractions ($\frac{2}{5}$), decimals ($0.4$), or percentages ($40%$). You must be able to convert fluently between these three representations without calculation errors.

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Sample Space ($S$) & Theoretical Probability

1. Key Definitions

• Random Experiment: A repeatable process with uncertain outcomes (e.g., spinning a spinner, selecting a marble from a bag).
• Sample Space ($S$): The set of all possible outcomes of a random experiment. The total number of outcomes is denoted $n(S)$.
• Event ($E$): A specific subset of outcomes within the sample space that satisfies a given condition. The number of favorable outcomes is denoted $n(E)$.

2. Theoretical Probability Formula

When all outcomes in a sample space are equiprobable (equally likely to occur due to physical symmetry), the theoretical probability of event $E$ is given by:

$P(E) = \frac{n(E)}{n(S)} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes in sample space}}$

Conceptual Example: Standard 52-Card Deck

A standard deck of playing cards contains $n(S) = 52$ cards divided into 4 suits (Hearts, Diamonds, Clubs, Spades) of 13 cards each.

• Event $A$ = Drawing a Heart: $n(A) = 13 \implies P(A) = \frac{13}{52} = \frac{1}{4} = 0.25 = 25%$.
• Event $B$ = Drawing a King: $n(B) = 4 \implies P(B) = \frac{4}{52} = \frac{1}{13} \approx 0.0769 = 7.69%$.
• Event $C$ = Drawing a Red Face Card (J, Q, K of Hearts/Diamonds): $n(C) = 6 \implies P(C) = \frac{6}{52} = \frac{3}{26} \approx 0.1154 = 11.54%$.

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Experimental Probability & Empirical Data

While theoretical probability predicts outcomes based on ideal mathematical conditions, experimental probability (also called empirical probability) measures what actually occurs during real-world testing or data collection.

Experimental Probability Formula

$P_{\text{exp}}(E) = \frac{f}{N} = \frac{\text{Observed frequency of event } E}{\text{Total number of conducted trials } N}$

Where:

• $f$ is the count of trials in which event $E$ occurred.
• $N$ is the total number of experimental trials performed.

Comparing Theoretical and Experimental Frameworks

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The Law of Large Numbers

The bridge between the two is the Law of Large Numbers (LLN). The blueprint lists solving problems involving probability rather than the theorem by name, but the theoretical-versus-experimental distinction it explains is exactly the reasoning probability items reward. This theorem explains the connection between theoretical expectations and empirical observations.

Law of Large Numbers Definition: As the total number of independent experimental trials $N$ increases toward infinity ($N \to \infty$), the experimental probability $P_{\text{exp}}(E)$ converges toward the true theoretical probability $P(E)$.

$\lim_{N \to \infty} P_{\text{exp}}(E) = P(E)$

Empirical Trial Simulation: Coin Tossing Example

Consider tossing a fair coin ($P(\text{Heads}) = 0.50$). The table below illustrates how experimental results stabilize as trial counts scale up:

Classroom & Pedagogical Application

In Ontario elementary and secondary classrooms, teachers use virtual probability simulators (e.g., spinning wheels or dice rollers) to demonstrate the Law of Large Numbers. Small sample sizes ($N = 10$) produce significant variability, leading students to falsely suspect bias. Pooling data across the entire class ($N = 500$) demonstrates empirical convergence toward the theoretical baseline.

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Complementary Events: $P(\text{not } E) = 1 - P(E)$

For any event $E$, its complement (denoted $E'$ or $\text{not } E$) comprises all outcomes in the sample space $S$ that are not contained in $E$.

Because event $E$ and its complement $E'$ are mutually exclusive and collectively cover the entire sample space:

$n(E) + n(E') = n(S)$

Dividing all terms by $n(S)$ yields the fundamental Complement Rule:

$P(E) + P(E') = 1 \implies P(\text{not } E) = 1 - P(E)$

Strategic Utility on the MPT

Calculating $P(\text{not } E)$ using $1 - P(E)$ is often dramatically faster than counting all non-favorable outcomes individually, especially when solving "at least one" probability problems.

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Worked Step-by-Step Problem Walkthroughs

Problem 1: Theoretical Probability with Multi-Property Spinners

Problem Statement: A game spinner is divided into 12 equal, congruent sectors numbered from 1 to 12. A player spins the arrow once.

1. Calculate the theoretical probability of landing on a prime number.
2. Calculate the probability of landing on a sector that is not a multiple of 4.
3. Express both probabilities as simplified fractions, decimals rounded to three places, and percentages.

Step-by-Step Solution:

1. Identify the Sample Space: $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\} \implies n(S) = 12$

2. Part 1: Calculate $P(\text{prime})$:

   • Identify prime numbers in $S$ (note: 1 is not prime): $\text{Primes} = \{2, 3, 5, 7, 11\} \implies n(\text{prime}) = 5$
   • Apply formula: $P(\text{prime}) = \frac{n(\text{prime})}{n(S)} = \frac{5}{12} \approx 0.417 = 41.7\%$

3. Part 2: Calculate $P(\text{not a multiple of 4})$ using Complement Rule:

   • First, find $n(\text{multiple of 4})$: $\text{Multiples of 4} = \{4, 8, 12\} \implies n(\text{multiple of 4}) = 3$
   • Compute $P(\text{multiple of 4})$: $P(\text{multiple of 4}) = \frac{3}{12} = \frac{1}{4} = 0.25$
   • Apply complement rule: $P(\text{not multiple of 4}) = 1 - P(\text{multiple of 4}) = 1 - \frac{1}{4} = \frac{3}{4} = 0.75 = 75.0\%$

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Problem 2: Experimental Data & Population Projection

Problem Statement: A Grade 8 class conducts an experiment by drawing color chips from an opaque container, recording the result, and returning the chip. Over 250 trials, the students record the following frequencies: Red = 65, Blue = 110, Yellow = 45, Green = 30.

1. Determine the experimental probability of drawing a Blue chip.
2. If the container actually holds a total of 50 chips, estimate how many Blue chips are inside the container.

Step-by-Step Solution:

1. Calculate Experimental Probability $P_{\text{exp}}(\text{Blue})$: $N = 250, \quad f_{\text{blue}} = 110$ $P_{\text{exp}}(\text{Blue}) = \frac{110}{250} = \frac{11}{25} = 0.44 = 44.0\%$

2. Estimate Expected Population Count in Container:

   • Multiply total container chips ($50$) by the experimental relative frequency ($0.44$): $\text{Estimated Blue Chips} = 50 \times 0.44 = 22$
   • Conclusion: Based on empirical sampling, approximately 22 out of 50 chips in the container are blue.