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
“Weather forecasts, insurance and cricket predictions all run on probability.”
A frequent error is assuming P(A|B) equals P(B|A). In reality, P(A|B) is the probability of A given that B has occurred, equal to P(A∩B)/P(B).
Memory trick: conditioning direction matters — Bayes' theorem flips it correctly.
“Weather forecasts, insurance and cricket predictions all run on probability.”
The addition rule. Use it when any two events.
P(A∪B) = P(A)+P(B)−P(A∩B)
“Weather forecasts, insurance and cricket predictions all run on probability.”
Conditional probability. Use it when P(B) > 0.
P(A|B) = P(A∩B)/P(B)
“Weather forecasts, insurance and cricket predictions all run on probability.”
Independent events don't influence each other's probability; mutually exclusive events can't co-occur (so they're dependent).
“Weather forecasts, insurance and cricket predictions all run on probability.”
These conditional probabilities ask different questions and are generally unequal — Bayes' theorem relates them.
“Weather forecasts, insurance and cricket predictions all run on probability.”
Mutually exclusive events cannot both occur, so they are dependent, not independent.
“A pilot adds wind and engine thrust as vectors to know where the plane actually goes.”
A frequent error is expecting the dot product to be a vector. In reality, a·b = |a||b|cosθ gives a scalar and is zero when the vectors are perpendicular.
Memory trick: dot product → scalar; it measures how much two vectors align.
“A pilot adds wind and engine thrust as vectors to know where the plane actually goes.”
A frequent error is expecting the cross product to be a scalar. In reality, a×b = |a||b|sinθ·n̂ gives a vector perpendicular to both a and b.
Memory trick: cross product → vector; its length is the parallelogram's area.
“A pilot adds wind and engine thrust as vectors to know where the plane actually goes.”
The components that fix a line's or vector's direction in 3D space.
Memory trick: direction cosines are direction ratios divided by the magnitude.
“A pilot adds wind and engine thrust as vectors to know where the plane actually goes.”
A frequent error is confusing direction ratios with direction cosines (which are normalised). In reality, the components that fix a line's or vector's direction in 3D space.
Memory trick: direction cosines are direction ratios divided by the magnitude.
“A pilot adds wind and engine thrust as vectors to know where the plane actually goes.”
Dot product (scalar). Use it when θ is the angle between the vectors.
a·b = |a||b|cosθ
“A pilot adds wind and engine thrust as vectors to know where the plane actually goes.”
Magnitude of the cross product (vector). Use it when θ is the angle between the vectors.
|a×b| = |a||b|sinθ