Probability & Statistics

Probability & Statistics

Concept

Statistics is about summarizing and understanding data. When someone says “the average score in the exam was 72,” that’s a statistic — a single number that summarizes a whole dataset. The three main measures of central tendency (where the data clusters) are Mean, Median, and Mode.

Mean is what most people mean by “average” — sum divided by count. It’s the most common measure but gets pulled around by extreme values (outliers).

Median is the middle value when you sort the data. Half the values are above it, half below. The median doesn’t budge when outliers swing extreme — it’s more stable.

Mode is the most frequent value. A dataset can have no mode (all values unique), one mode (unimodal), or two+ modes (bimodal/multimodal).

Probability answers: “How likely is this to happen?” Probability is always between 0 and 1, where 0 means impossible and 1 means certain. The classical approach: P(Event) = Number of favorable outcomes / Total number of possible outcomes.

Permutations vs Combinations — this is where students often get confused. Permutations (nPr) count arrangements where ORDER MATTERS. Combinations (nCr) count groups where ORDER DOESN’T MATTER. Example: selecting a captain and vice-captain from 5 students is a permutation (order matters: A-captain, B-vice is different from B-captain, A-vice). Selecting any 3 students to form a committee is a combination (A+B+C is the same as C+B+A).

Key Formulas

Formula	Use
Mean = Σx / n	Average of raw data
Median = middle value (sorted)	Central value, outlier-resistant
Mode = most frequent value	Most common occurrence
P(event) = Favorable / Total	Classical probability
P(complementary) = 1 - P(event)	“At least one” problems
P(A or B) = P(A) + P(B) - P(A∩B)	Union of two events
P(A∩B) = P(A) × P(B)	Independent events
nPr = n! / (n - r)!	Arrangements (order matters)
nCr = n! / [r!(n - r)!]	Selections (order doesn’t matter)

Worked Example

Q: From a deck of 52 cards, what is the probability of drawing an Ace?

Step 1: Number of favorable outcomes (Aces) = 4 Step 2: Total possible outcomes (cards) = 52 Step 3: P(Ace) = 4/52 = 1/13

Answer: 1/13

Common Errors

• Confusing nPr and nCr → If the problem mentions “arrangement,” “order,” or “sequence” — it’s nPr. If it says “selection,” “group,” or “team” — it’s nCr.
• Forgetting to simplify fractions → Always reduce probability fractions to lowest terms (4/52 = 1/13)
• Probability > 1 or < 0 → Impossible! Probability is always between 0 and 1. If you get >1, something went wrong.