8.2 Chance and Uncertainty

Chance is a number between 0 and 1

Chance on NBT Quantitative Literacy is a way of measuring uncertainty in campus-scale stories. The Centre for Educational Assessments describes this subdomain as quantifying the chance of uncertain events from empirically derived data, including the idea of a random sample, and as writing probability as a number between 0 and 1. Zero means the event cannot happen on the information given. One means it must happen. Values such as 0.25, 1/4, and 25% name the same chance; QL options sometimes mix decimals and percentages to see whether you notice the scale.

Start with empirical chance, which is just a careful fraction. Marula University's campus clinic recorded 80 bookings on a Monday: 12 influenza reviews, 28 vaccination slots, and 40 sports-physio follow-ups. If one booking is drawn at random from that list, the probability it is a vaccination slot is 28/80 = 0.35. The probability it is influenza is 12/80 = 0.15. The probability it is not physio is (12 + 28)/80 = 0.50. You do not need a formula named "complement" if you can see the leftover count, but you do need the same denominator — 80 bookings, not 80 students in the faculty, and not 28 because vaccination "feels common."

Equally likely outcomes are a modelling choice, not a decoration. If a tutorial coordinator has six identical groups and assigns each student by a fair draw among those six labels, P(group C) = 1/6 ≈ 0.167. The assumption fails as soon as the groups are different sizes and assignment is to a seat. Four residence names are not four equally likely beds. Suppose a late applicant is assigned uniformly at random to a remaining bed, and the remaining beds are 80, 80, 80, and 40 in four blocks A, B, C and D. Total beds = 280. Probability of block D is 40/280 = 1/7, not 1/4. Treating "four names" as "four equal chances" is one of the most common QL errors.

Random-sample language on QL is practical. "A student is picked at random from the 400 first-years" means each of those 400 had the same chance to be named. It does not mean each faculty had the same chance, because Commerce is larger than Education. If the sentence says "a Health Sciences student is picked at random," the sample space has already been narrowed to 120 people. Changing the group in the sentence changes the denominator. Tickets, beds, bookings, and faculties are different objects; name the object before you divide.

Two-way tables are the workhorse for simple compound events. The same 400 first-years were classified by faculty and by whether they use the campus shuttle from the Polokwane taxi rank.

AND is the overlapping cell. Probability a randomly chosen first-year is in Health Sciences and uses the shuttle is 48/400 = 0.12, not 48/120 (that would be among Health students only) and not 120/400 (that is only the faculty). OR for these categories is "Health or shuttle or both": add the Health total to the shuttle total, then subtract the overlap so it is not counted twice: (120 + 158 − 48)/400 = 230/400 = 0.575. If the question says "Health and Education," the events cannot both happen for one student; that AND is 0. Mutually exclusive faculty labels are a context fact, not a theorem you have to prove.

A tree of the same numbers is a picture of those cells. From 400, three faculty branches (120, 80, 200); from Health, 48 and 72; and so on. On QL you mostly read along a path and divide the path count by the starting total. That is different from MAT probability trees, where you are often required to multiply branch probabilities, apply independence, and check that complementary branches sum to 1 as a formal rule. MAT also uses Venn diagrams to shade unions and intersections with set notation. QL may show a tree or a table; it is still asking for a chance in context, not for a proof that P(A ∪ B) = P(A) + P(B) − P(A ∩ B). You can use that identity as a private shortcut. You should not expect QL to test Venn-region algebra the way the MAT paper can.

Compound events stay simple on QL: two attributes from a table, or two stages that the passage already counted. Example: probability a random first-year is in Commerce and does not use the shuttle is 110/400 = 0.275. Example: if a shuttle user is selected at random, the probability they are in Education is 20/158 ≈ 0.127. The second sentence changed the sample to the 158 shuttle users. That is still table-reading, not Bayes' theorem by name.

Keep the 0–1 scale honest. A probability of 1.2 is impossible. A statement that an event is "likely" is not a number until the table supplies a fraction. "Almost certain" is not 1 unless the leftover count is zero. Conversely, a very small positive probability (2 remaining isolation beds out of 280, so 2/280 ≈ 0.007) is not 0; the event is rare, not impossible.

Compared with QL chance items, MAT probability can ask counting techniques, independent events in abstract sample spaces, and Venn calculations with three sets. If you find yourself drawing overlapping circles and solving for an unknown region with algebra, you have slipped into MAT method. Return to the table, name the denominator, and write a number between 0 and 1. Slow down on the denominator.