8.2 Chance & Uncertainty

Why chance and uncertainty appear on QL

Official NBT National Reports score six Quantitative Literacy subdomains, one of which is Chance and uncertainty. Writers who only practise tables and arithmetic still leave marks on the table when a stem asks how likely an event is, what a relative frequency means, or how to combine stages on a tree.

On AQL you will not need advanced combinatorics. You will need to:

• Translate everyday chance language into a sensible ordering or a numeric probability
• Compute empirical probabilities from counts
• Use equally likely outcome models when the stem says outcomes are fair / random / equally likely
• Read or build a simple tree for sequential stages
• Interpret uncertainty in context without overclaiming

No calculator is allowed, so keep fractions friendly and convert to decimals or percents only when the options demand it.

Chance language on a scale

Everyday words map onto the probability interval from impossible to certain:

Probabilities live on [0, 1] as decimals (or equivalent fractions). The same values appear as percentages from 0% to 100%. Convert with ×100 or ÷100: 0.25 ↔ 25%, 3/5 = 0.6 ↔ 60%.

If a stem says a campus Wi-Fi outage is “unlikely but possible,” an option claiming probability 1 is wrong, and an option claiming probability 0 is also wrong. If two events are described as “equally likely,” their probabilities should match (often each 1/2 when there are exactly two outcomes and nothing else can occur).

Complementary events

If event A and “not A” partition all possibilities, then P(A) + P(not A) = 1. So if the empirical chance that a shuttle arrives full is 0.25, the chance it is not full is 0.75. Complementary thinking often saves a longer tree calculation.

Empirical probability (relative frequency)

When data already exist, the natural estimate is:

Empirical probability ≈ (number of favourable outcomes) ÷ (number of trials / observations)

Worked example: load-shedding and tutorials

Over 80 scheduled tutorial slots in a term, load-shedding forced cancellation of 24 slots.

• Empirical P(cancellation) = 24 / 80 = 0.3 = 30%
• Empirical P(tutorial runs) = 56 / 80 = 0.7 = 70%

Meaning in context: based on this term’s record, about three in ten slots were cancelled. That does not mean the next slot is “certain” to run or “certain” to cancel. Relative frequency describes the sample of 80, not a guarantee about one future Wednesday.

Sample size matters

Compare two residence kitchens:

• Kitchen A: 2 of 5 surveyed evenings had no hot water → 2/5 = 40%
• Kitchen B: 40 of 200 surveyed evenings had no hot water → 40/200 = 20%

Kitchen A’s 40% looks “worse,” but it rests on only five evenings. Kitchen B’s 20% is estimated from a much larger sample. QL stems often ask which claim is better supported, or warn you not to treat a tiny sample’s percentage as decisive. Larger samples usually give more stable relative frequencies; tiny samples swing wildly.

Another empirical read: clinic visits

A campus clinic logged 150 walk-ins in a week; 45 were for respiratory symptoms.

• P(respiratory | walk-in in this week’s data) = 45 / 150 = 0.3 = 30%
• If options show 45% or 150%, someone misread the fraction as “45 out of 100” or used the denominator as a percent.

Always name the reference class: “of these 150 walk-ins,” not “of all students forever.”

Equally likely outcomes

When a stem states that outcomes are equally likely (fair coin, fair die, random digit, names drawn at random from a complete list), use:

P(event) = (number of favourable outcomes) ÷ (number of possible outcomes)

Worked example: student card last digit

Suppose a last digit 0–9 is treated as equally likely.

• P(digit is even) = outcomes {0,2,4,6,8} → 5/10 = 1/2 = 0.5
• P(digit is 7) = 1/10 = 0.1
• P(digit is at least 8) = {8,9} → 2/10 = 0.2

Worked example: tutorial group draw

A practical class has 12 students; 3 are from Residence East. One name is drawn at random for a demo.

• P(East) = 3/12 = 1/4 = 0.25
• P(not East) = 9/12 = 0.75

If two students are drawn without replacement, the second-stage denominator shrinks — that is a cue to switch to a tree or a careful two-step fraction, not to reuse 3/12 blindly.

Tree diagrams: more than one fully worked path set

A tree splits at each stage. Multiply probabilities along a path. Add probabilities of mutually exclusive paths that produce the same overall result.

Worked tree 1 — lab equipment free

Students book either a morning slot (0.7) or an afternoon slot (0.3).

• If morning: equipment free with probability 0.9 (busy with 0.1).
• If afternoon: equipment free with probability 0.4 (busy with 0.6).

Probability equipment is free:

• Morning and free: 0.7 × 0.9 = 0.63
• Afternoon and free: 0.3 × 0.4 = 0.12
• Total free: 0.63 + 0.12 = 0.75

Probability busy = 1 − 0.75 = 0.25, which matches 0.7 × 0.1 + 0.3 × 0.6 = 0.07 + 0.18 = 0.25.

Worked tree 2 — printing before a deadline

A student either uses the library queue (0.6) or the faculty lab (0.4).

• Library: print succeeds 0.85, fails 0.15.
• Faculty lab: print succeeds 0.55, fails 0.45.

P(success) = 0.6 × 0.85 + 0.4 × 0.55 = 0.51 + 0.22 = 0.73.

P(fail) = 0.6 × 0.15 + 0.4 × 0.45 = 0.09 + 0.18 = 0.27, and 0.73 + 0.27 = 1.

Worked tree 3 — counts instead of decimals

Of 200 survey respondents, 120 are on main campus and 80 are on the satellite campus. Among main-campus respondents, 90 prefer evening lectures; among satellite respondents, 20 prefer evening lectures.

• P(main then evening) = (120/200) × (90/120) = 90/200 = 0.45
• P(satellite then evening) = (80/200) × (20/80) = 20/200 = 0.10
• P(evening) = 0.45 + 0.10 = 0.55

Notice the 120 cancels in the first path: you are really counting the 90 evening-preferring main-campus students out of 200. Trees and two-way tables agree when you stay disciplined about the sample.

Uncertainty in data and context

Chance items are often wrapped in a story: weather delaying fieldwork, taxi arrival, NSFAS allowance timing, or whether a residence generator covers an evening study session. Read the story for:

1. What is random? (which event’s probability is asked)
2. What information is given? (counts, equally likely claim, or branch probabilities)
3. What would be an overclaim? (“will definitely,” “proves,” “always”) when only a frequency is known

A 70% empirical success rate for online registration attempts means registration often works in that sample — not that your single attempt tonight is certain.

Common traps

Quick self-check before you pick an option

1. Is the stem asking for a word (likely/unlikely) or a number?
2. Are outcomes equally likely, or must I use given frequencies/branch probs?
3. If there are stages, did I multiply along and add across?
4. Does my answer sit in [0, 1] (or 0%–100%)?
5. Did I avoid upgrading an estimate into a certainty claim?

Chance and uncertainty rewards precise language as much as arithmetic. Treat every probability as a controlled claim about a defined set of outcomes — then the MCQ options become much easier to eliminate.