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Counting & Probability · Recognition Drill

Counting & Probability · 35 stems · 8s per card
35 contest stems. Classify each one — don't compute it. 5-second budget (8 in coach mode). Counting and probability share the same toolkit, so this drill mixes them.
Year-8 notation glossary — read this first
(nk)\binom{n}{k}
Number of ways to choose kk items from nn when order doesn't matter. (52)=10\binom{5}{2} = 10.
n!n! (factorial)
n×(n−1)×⋯×1n \times (n-1) \times \dots \times 1. 5!=1205! = 120, 0!=10! = 1.
P(A)P(A)
Probability that event AA happens — a number between 00 and 11.
P(A∣B)P(A \mid B)
Probability of AA GIVEN that BB has happened (conditional probability).
P(A∩B)P(A \cap B)
Probability that BOTH AA AND BB happen.
P(A∪B)P(A \cup B)
Probability that AT LEAST ONE of AA, BB happens.
E[X]E[X]
Expected value (long-run average) of the random variable XX.
xˉ\bar x
The mean (arithmetic average) of a set of numbers.
Technique deck — what the button labels mean
Complementary counting
Count the OPPOSITE event, subtract from the total. Use when 'at least one' or 'not all' appears.
Stars and bars
Number of ways to distribute n identical items into k bins: $\binom{n+k-1}{k-1}$.
Inclusion-exclusion (PIE)
$|A \cup B| = |A| + |B| - |A \cap B|$ (and the multi-set generalisation).
Pigeonhole principle
If $n+1$ items in $n$ boxes, some box has ≥ 2 items. Use for 'must share' problems.
Casework
Partition the problem into disjoint scenarios, count each, add up.
Product rule
Independent choices multiply: $a \times b \times c$ outcomes.
Sum rule
Disjoint cases add: count A or count B, no overlap.
Permutations
Ordered arrangements: $n!$ for all, $n!/(n-r)!$ for $r$ out of $n$.
Combinations
Unordered selections: $\binom{n}{r}$.
Multinomial (permutations with repetition)
$n! / (n_1! n_2! \cdots n_k!)$ when some items repeat.
Circular permutations
Arrangements around a circle: $(n-1)!$ (fix one seat to remove rotation).
Bijection / lattice paths
Map the count to a well-known one (e.g., lattice paths = $\binom{m+n}{m}$).
Generating functions
Encode counts as polynomial coefficients; products track combined choices.
Basic probability
$P = $ favourable outcomes / total outcomes (equally likely).
Independent events
$P(A \cap B) = P(A) \cdot P(B)$ — only when $A, B$ don't influence each other.
Conditional probability
$P(A \mid B) = P(A \cap B) / P(B)$ — probability of A given B happened.
Bayes' theorem
$P(A \mid B) = P(B \mid A) P(A) / P(B)$ — flip the condition.
Geometric probability
$P = $ favourable area / total area (continuous, uniform setting).
Expected value (definition)
$E[X] = \sum_i p_i x_i$ — probability-weighted average outcome.
Linearity of expectation
$E[X + Y] = E[X] + E[Y]$ regardless of dependence. Use with indicator variables.
Geometric distribution
Trials until first success. Expected number of trials $= 1/p$.
Derangements / fixed-point-free
Permutations with NO fixed point; count via inclusion-exclusion.
Mean
Arithmetic average: $\sum x_i / n$.
Median
Middle value of a sorted list (or average of two middle values for even n).
Variance
$\sigma^2 = \sum (x_i - \bar x)^2 / n$ — average squared deviation from the mean.
Mode
Most-frequently-occurring value in a dataset.