10.1 Theoretical vs. Experimental Probability, Complementary Events, and Expected Frequency

The Probability Scale and Equivalent Forms

Probability measures the likelihood that a specified event will occur on a numerical scale from $0$ to $1$. An impossible event has a probability of $0$ (or $0\%$), an equally likely or "even chance" event has a probability of $0.5$ (or $\frac{1}{2}$ or $50\%$), and a certain event has a probability of $1$ (or $100\%$). No valid probability can ever be negative or greater than $1$.

In a random experiment, the sample space is the complete set of all possible individual outcomes. When a sample space is partitioned into mutually exclusive and exhaustive categories—meaning every trial produces exactly one category—the sum of all category probabilities must equal $1$ (or $100\%$).

LANTITE questions frequently require converting probabilities across three equivalent representations:

• Simplified fractions: Divide numerator and denominator by their highest common factor (for example, $\frac{15}{40} = \frac{3}{8}$).
• Decimals: Divide the numerator by the denominator (for example, $3 \div 8 = 0.375$).
• Percentages: Multiply the decimal or fraction by $100\%$ (for example, $0.375 \times 100\% = 37.5\%$).

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Theoretical Probability and Complementary Events

When the physical symmetry or random selection rules of a scenario guarantee that every elementary outcome in the sample space is equally likely (such as rolling a fair die, drawing a student name from a thoroughly mixed hat, or spinning a uniform sector wheel), the theoretical probability of an event $E$ is:

$$P(E) = \frac{\text{Number of favourable outcomes}}{\text{Total number of equally likely outcomes}}$$

The complement of event $E$, written $E'$ or "$\text{not } E$", consists of every outcome in the sample space where $E$ does not occur. Because an event and its complement together cover the entire sample space without overlap:

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

The complement rule provides a powerful shortcut for "at least one" problems. Whenever a question asks for the probability that at least one item meets a condition, or that an outcome does not fail every time, the complement of "at least one" is "none":

$$P(\text{at least one}) = 1 - P(\text{none})$$

Worked Example: Classroom Token Draw and Missing Category Probability

A teacher places $8$ blue tokens, $5$ red tokens, $4$ green tokens, and $3$ gold tokens into an opaque bag ($8 + 5 + 4 + 3 = 20$ tokens in total). If one token is drawn at random:

• $P(\text{gold}) = \frac{3}{20} = 0.15 = 15\%$
• $P(\text{not gold}) = 1 - 0.15 = 0.85 = 85\%$ (equivalently, $\frac{8 + 5 + 4}{20} = \frac{17}{20}$)

Similarly, if a school library categorises all borrowed books into four mutually exclusive genres— Junior Fiction ($0.38$), Non-Fiction ($0.27$), Graphic Novels ($0.19$), and Senior Fiction—you can find the missing probability for Senior Fiction by subtracting the known probabilities from $1$: $P(\text{Senior Fiction}) = 1 - (0.38 + 0.27 + 0.19) = 1 - 0.84 = 0.16$ (or $16\%$). If $250$ books are borrowed on Friday, the expected number of Senior Fiction loans is $250 \times 0.16 = 40$ books.

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Mutually Exclusive vs. Non-Mutually Exclusive Events

When calculating the probability that event $A$ or event $B$ occurs (where "or" in mathematics always means at least one of the two), you must first check whether the two events can happen together on the same single selection.

1. Mutually Exclusive (Disjoint) Events: Events $A$ and $B$ cannot occur simultaneously on a single trial, so $P(A \text{ and } B) = 0$. For example, a randomly selected student cannot be enrolled in both Year 7 and Year 9 at the same time. For mutually exclusive events, simply add the individual probabilities:

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

1. Non-Mutually Exclusive (Overlapping) Events: Events $A$ and $B$ can occur together on the same selection, so $P(A \text{ and } B) > 0$. For example, a student can belong to both the school choir and the debating team. Simply adding $P(A) + P(B)$ counts the overlapping students twice, so you must subtract the overlap once:

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

Worked Example: Overlapping Subject Enrolments

In a cohort of $120$ Year 8 students, $42$ study Japanese, $36$ study Drama, and $18$ study both Japanese and Drama. If one Year 8 student is chosen at random:

• The number of students studying Japanese or Drama (or both) is $42 + 36 - 18 = 60$.
• Therefore, $P(\text{Japanese or Drama}) = \frac{60}{120} = \frac{1}{2} = 0.50 = 50\%$.
• The number of students studying Japanese only (without Drama) is $42 - 18 = 24$, so $P(\text{Japanese only}) = \frac{24}{120} = 0.20 = 20\%$.
• By the complement rule, the number of students studying neither subject is $120 - 60 = 60$, giving $P(\text{neither}) = 1 - 0.50 = 0.50$.

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Experimental Probability and the Law of Large Numbers

In many real-world educational settings—such as estimating the chance that a library borrower returns a book early or testing an asymmetrical 3D-printed spinner—outcomes are not equally likely by symmetry. Instead, we estimate probability from observed data using experimental (empirical) probability, also known as relative frequency:

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

Because of natural random variation, experimental probability over a small number of trials often differs from the theoretical probability. For example, flipping a fair coin $10$ times ($P(\text{Heads}) = 0.50$) might easily yield $7$ Heads ($P_{\text{exp}} = 0.70$). However, the Law of Large Numbers guarantees that as the total number of independent trials $N$ grows larger, the experimental relative frequency converges toward the true underlying probability. When a LANTITE question presents experimental results from several student groups of different sample sizes, the most reliable estimate of the true probability is obtained from the largest sample size $N$—or best of all, by pooling (combining) the observed counts and total trials across all groups.

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Expected Frequency and School Fundraising Calculations

Once you know the probability $P(E)$ of an event—whether from a theoretical model or a large empirical record—the expected frequency (the expected number of occurrences) across $N$ trials or a cohort of $N$ students is:

$$E = N \times P(E)$$

Expected frequency represents a long-run mathematical average rather than an exact guarantee for a single run.

Worked Example: School Fete Raffle Return

A primary school P&C sells $500$ raffle tickets at $4 per ticket, collecting $500 \times 4 = 2{,}000$ dollars in gross ticket sales. The prize pool consists of $1$ first prize valued at $450, $2$ second prizes valued at $150 each, and $5$ third prizes valued at $30 each.

• Probability of a single ticket winning any prize: There are $1 + 2 + 5 = 8$ winning tickets out of $500$, so $P(\text{win}) = \frac{8}{500} = \frac{2}{125} = 0.016 = 1.6\%$.
• Total prize cost: $450 + (2 \times 150) + (5 \times 30) = 450 + 300 + 150 = 900$ dollars.
• Net fundraising profit: $2{,}000 - 900 = 1{,}100$ dollars overall, which equals $\frac{1{,}100}{500} = 2.20$ dollars net profit per ticket sold.