Fundamentals
The core facts every aspirant should own — each a titled nugget with a real-world story, the concept in plain words, and a memory trick. Works even when the internet doesn't.
300 fundamentals
“Seeing a friend’s “online” badge (B) and asking the chance they’ll reply (A) uses P(A|B), not the reverse.”
P(A|B) is the probability that A occurs when B is known to have happened; mathematically P(A|B)=P(A∩B)/P(B) (P(B)>0). A common error is swapping A and B, giving P(B|A)=P(A∩B)/P(A). They match only in trivial zero-probability cases or when P(A)=P(B).
Memory trick: Condition → denominator: P(A|B)=P(A∩B) ÷ P(B).
“The chance of drawing a heart or a king equals P(heart)+P(king)−P(heart∧king).”
For any two events, P(A∪B)=P(A)+P(B)−P(A∩B). The subtraction removes the double-counted overlap. Forgetting the intersection overestimates the probability.
P(A∪B) = P(A)+P(B)−P(A∩B)
Memory trick: Add, then subtract overlap.
“In a bag, the chance of a red ball given you picked from the left half is (red∧left)/P(left).”
Conditional probability formula P(A|B)=P(A∩B)/P(B) holds only when P(B)>0; otherwise the expression is undefined. Always check the denominator first.
P(A|B) = P(A∩B)/P(B)
Memory trick: Denominator must be >0.
“Flipping two coins: each result doesn't change the other's chance (independent); picking a red card means you can't pick a black one in that same draw (mutually exclusive).”
Independent events satisfy P(A∩B)=P(A)·P(B). Mutually exclusive events have P(A∩B)=0, so P(A|B)=0 and they are dependent unless one event has zero probability. Exam trap: using the multiplication rule for exclusive events gives zero only when an event is impossible.
Memory trick: Indep → multiply; Exclusive → zero overlap.
“A doctor sees a positive test (B) and wonders about the disease chance (A); the reverse asks the chance a diseased patient tests positive.”
P(A|B)=P(A∩B)/P(B) requires P(B)>0; P(B|A)=P(A∩B)/P(A) requires P(A)>0. They differ in general. When both are defined, they are equal only if the events are disjoint (P(A∩B)=0) or their marginal probabilities are equal, P(A)=P(B).
Memory trick: Equal ⇔ same marginal or no overlap (P(A∩B)=0).
“If a dice shows 6, it cannot simultaneously show 1 – the events are mutually exclusive, not independent.”
Mutually exclusive events give P(A∩B)=0, so P(A|B)=0, showing they are dependent. Independence requires P(A∩B)=P(A)P(B). The myth fails because zero overlap equals independence only when one event’s probability is zero.
Memory trick: Zero overlap ≠ independent unless impossible.
“When you push a swing, the work you do equals the dot product of force and displacement.”
The dot product a·b = |a||b|cosθ is a scalar. It measures how much one vector projects onto the other; it becomes zero when the vectors are perpendicular. A common trap is to treat the result as a vector or to forget the cosine factor.
Memory trick: Dot = scalar; a dot is a point → one number.
“The torque you feel turning a wrench is the cross product of force and lever arm.”
The cross product a×b = |a||b|sinθ n̂ is a vector perpendicular to both a and b, with magnitude equal to the area of the parallelogram they span. Students often mistake it for a scalar or ignore the right-hand rule for its direction.
Memory trick: Cross = vector; X points out of the page (right-hand rule).
“A kite’s line direction in the sky is described by its direction ratios (dx, dy, dz).”
Direction ratios (l, m, n) are any set of numbers proportional to the components of a line’s direction vector. They define the line’s orientation in 3-D; any multiple represents the same direction. Remember they are not normalized.
Memory trick: DR are raw components; any multiple works.
“Aligning a satellite dish uses direction cosines, not raw direction ratios.”
Direction cosines (α, β, γ) are the normalized direction ratios: α = l/√(l²+m²+n²) etc., and satisfy α²+β²+γ²=1. Confusing them with raw ratios leads to wrong angle calculations.
Memory trick: Cosine = DR divided by its length.
“Two arrows pointing the same way have dot product = product of their lengths.”
a·b = |a||b|cosθ gives a scalar equal to the product of the magnitudes times the cosine of the angle between them. Use it to find the angle or projection; forgetting the absolute value signs or using degrees incorrectly is a frequent error.
a·b = |a||b|cosθ
Memory trick: Dot = |a||b|·cosθ.
“The cross product’s magnitude is the area of the parallelogram formed by the two side-vectors, like the floor you’d tile using two fence lengths at an angle.”
|a×b| = |a||b|sinθ gives the area of the parallelogram spanned by a and b. It vanishes when the vectors are collinear (θ=0 or π) or when either vector is the zero vector. In JEE, remember to include the anti-parallel and zero-vector cases.
|a×b| = |a||b|sinθ
Memory trick: Area = side·side·sinθ → zero if sinθ=0 or a=0 or b=0