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Probability: Basic revision notes

Every idea in the Probability: Basic chapter for JEE Main, with the usual mistakes and the fast methods. Topic 1 is open to everyone; sign in free to read all 16 and download the PDF.

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How we count: 7× asked means 7 past JEE Main questions in the PrepWiser bank needed that concept, and "last" is the latest year one appeared. ★ marks ideas asked 8 or more times.

Sample space and events

Classical probability
6× asked, last 2026
P(E)=n(E)n(S),0≤P(E)≤1,P(∅)=0, P(S)=1P(E)=\frac{n(E)}{n(S)},\qquad 0\le P(E)\le 1,\qquad P(\emptyset)=0,\ P(S)=1
Finite sample space of equally likely outcomes only. Count EE and SS the same way: both ordered or both unordered.
  • Not an ordered count over an unordered total. Keep one sample space. aa then bb from 1–100 without replacement: 9900 ordered pairs, of which 90+89+⋯+1=409590+89+\dots+1=4095 have a−b≥10a-b\ge10.
Unequal outcomes: add the weights
5× asked, last 2024
P(A)=∑ω∈AP(ω),∑ω∈SP(ω)=1P(A)=\sum_{\omega\in A}P(\omega),\qquad \sum_{\omega\in S}P(\omega)=1
Biased die or coin: outcomes are not equally likely, so add their probabilities instead of counting them.
  • Quick method. Fix the unknown from total =1=1 first. Head twice as likely as tail: P(H)=2/3P(H)=2/3. If S={ω1,ω2,… }S=\{\omega_1,\omega_2,\dots\} is infinite and P(ωn)=P(ωn−1)/2P(\omega_n)=P(\omega_{n-1})/2 for all n≥2n\ge2, the series gives P(ωn)=1/2nP(\omega_n)=1/2^n.
Complement, exclusive events, odds
2× asked, last 2023
P(A′)=1−P(A),A∩B=∅ ⇒ P(A∪B)=P(A)+P(B)P(A')=1-P(A),\qquad A\cap B=\emptyset\ \Rightarrow\ P(A\cup B)=P(A)+P(B)
Exhaustive: union is SS; exclusive and exhaustive give ∑P(Ei)=1\sum P(E_i)=1. Odds a:ba:b in favour: P(E)=a/(a+b)P(E)=a/(a+b).
  • Not only 0≤P(Ei)≤10\le P(E_i)\le1 for each event. Mutually exclusive events also need ∑P(Ei)≤1\sum P(E_i)\le1. That extra condition usually fixes one end of the range of the parameter.

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Practise Probability: Basic

13 more concepts in 5 topics

Addition theorem; Dice, coins and cards; Counting favourable cases; Equations, matrices and regions; and Independent trials. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Addition theorem

  • Addition theorem for two events3×
  • Addition theorem for three events3×

Dice, coins and cards

  • Dice6×
  • Coins and toss sequences3×

Counting favourable cases

  • Numbers with divisibility conditions9×
  • A.P. and G.P. triples3×
  • Arrangements and words4×
  • Urns: drawing without replacement4×
  • Random functions, subsets and boxes6×

Equations, matrices and regions

  • Quadratic with random coefficients6×
  • Matrices and systems with random entries6×
  • Geometric probability and boards5×

Independent trials

  • Product rule for independent trials11×

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Practise Probability: Basic

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