11.2 Basic Probability and Data Interpretation

11.2 Basic Probability and Data Interpretation

Quick Answer: Simple probability is $P(E) = \dfrac{\text{number of favorable outcomes}}{\text{number of equally likely outcomes}}$. Data-interpretation items ask you to read a table or chart correctly, then often convert a count into a fraction, decimal, or percent. Together, these skills turn raw displays into exam points.

USTET Mathematics includes Statistics topics from the Grade 11 curriculum: chance in everyday experiments and sense-making with organized data. You will see dice, coins, cards, spinner sectors, student surveys, and enrollment or score tables. The arithmetic is usually easy; the danger is miscounting the sample space or reading the wrong cell of a table under time pressure.

Probability Language

• An experiment is a process with uncertain results (rolling a die, drawing a name).
• An outcome is a single possible result (rolling a 4).
• The sample space $S$ is the set of all possible outcomes.
• An event $E$ is a subset of the sample space (rolling an even number: ${2, 4, 6}$).

If every outcome in $S$ is equally likely,

$P(E) = \frac{n(E)}{n(S)}.$

Probabilities satisfy $0 \leq P(E) \leq 1$. Impossible events have probability 0; certain events have probability 1. You may be asked for a fraction in lowest terms, a decimal, or a percent — match the form used in the options.

Classic fair die. Sample space size 6. $P(\text{even}) = 3/6 = 1/2$. $P(\text{at least 5}) = P(5\text{ or }6) = 2/6 = 1/3$.

Fair coin twice. Outcomes: HH, HT, TH, TT (four equally likely). $P(\text{exactly one head}) = 2/4 = 1/2$. $P(\text{two heads}) = 1/4$. Listing the sample space prevents double-counting mistakes.

Complement and Simple Combinations

The complement of event $E$, written $E'$ or $E^c$, is “$E$ does not happen.”

$P(E') = 1 - P(E).$

If $P(\text{rain}) = 0.3$, then $P(\text{no rain}) = 0.7$. Complement thinking is fastest when a stem asks for “not all,” “none,” or “at least one” after you can find the easier opposite event.

For mutually exclusive events that cannot occur together (rolling a 2 and rolling a 5 on one roll), add probabilities: $P(A \text{ or } B) = P(A) + P(B)$. For events that can overlap, you would subtract the intersection — USTET usually keeps overlaps obvious or avoids them.

For two independent events (coin and die, two draws with replacement), multiply: $P(A \text{ and } B) = P(A) \cdot P(B)$. Example: $P(\text{heads and a 6}) = (1/2)(1/6) = 1/12$.

Without replacement, outcomes are dependent. If a bag has 3 red and 2 blue marbles and you draw two without putting the first back, $P(\text{both red}) = (3/5)(2/4) = 6/20 = 3/10$. Update the denominator (and numerator) after each draw.

Empirical Probability from Data

When a table gives observed counts, probability is estimated by relative frequency:

$P(E) \approx \frac{\text{frequency of } E}{\text{total frequency}}.$

If 40 students chose STEM programs and 10 chose non-STEM in a survey of 50, the empirical probability a randomly selected surveyed student chose STEM is $40/50 = 0.8 = 80%$.

Reading Tables

Before computing, lock down:

1. Row and column headers — what does each label measure?
2. Totals — row totals, column totals, and grand total; verify they agree if both are printed.
3. The exact question — “how many,” “what percent of all,” and “what percent of those who…” use different denominators.

Two-way table example.

• How many STEM students need review? 12 (direct cell).
• What fraction of all students passed? $46/80 = 23/40$.
• Given that a student is ABM, probability they passed? $18/40 = 9/20$ (use the ABM row total, not 80).

Conditional wording (“given that,” “among those who”) is the #1 table trap: wrong denominator.

Bar Graphs, Line Graphs, and Pie Charts

Bar graphs compare categories. Read bar heights against the scale; watch for broken axes or scales that do not start at zero (less common on entrance exams, but always check the numbers printed).

Line graphs show change over time (months, school years). Focus on increases, decreases, peaks, and differences between two years — not on inventing a story the graph does not support.

Pie charts show parts of a whole. A sector labeled 25% of 200 students represents $0.25 \times 200 = 50$ students. If two sectors are 30% and 45%, together they are 75% of the whole. Sectors should total 100%; if options imply totals like 110%, reread the legend.

Percent, Fraction, and Decimal Fluency

Data items bounce among forms. Memorize quick conversions: $1/2 = 0.5 = 50%$, $1/4 = 0.25 = 25%$, $1/5 = 0.2 = 20%$, $3/8 = 0.375 = 37.5%$. When a chart gives percents and a stem gives a total population, multiply. When a stem gives raw counts and options are percents, divide and convert.

Worked chain. A pie chart shows that 35% of 240 examinees answered a Mental Ability item correctly. Number correct: $0.35 \times 240 = 84$. If 84 is also “favorable” for a follow-up probability among all examinees, $P(\text{correct}) = 84/240 = 0.35$, which matches the chart — a useful sanity check.

Avoiding Misreads

• Do not confuse number with percent in options.
• Do not compare bar heights across graphs that use different scales.
• Do not assume a steeper line segment means a larger percent change without checking actual values.
• For probability with “or,” clarify whether outcomes overlap.
• Reduce fractions only after you confirm numerator and denominator; premature canceling of digits (canceling 6s in 16/64 incorrectly) creates wrong choices that look tidy.

Probability and data interpretation reward slow reading of the display and fast arithmetic afterward. On USTET, spend your seconds identifying $n(E)$ and $n(S)$ — or the correct table cell — then compute once with care.