Probability — JEE Mains Mathematics

Formulas (2)

• Selections: counting with combinations · Classical probability
  $$P(E)=\frac{n(E)}{n(S)}$$
• Ordered outcomes: words, sequences and orders · Order by symmetry
  $$P(\text{given relative order of }k\text{ items})=\frac{1}{k!}$$

Watch out for (2)

• Same sample space in both counts
  → Selections: counting with combinations
• Include non-integer ratios
  → Ordered outcomes: words, sequences and orders

Formulas (2)

• Dice and weighted outcomes · Independent throws
  $$P(\text{result})=\sum_{\text{favourable }(a,b)}P(a)\,P(b)$$
• Numbers with divisibility or digit properties · Euler's function
  $$\varphi(N)=N\prod_{p\mid N}\left(1-\frac1p\right)$$

Watch out for (2)

• Ordered pairs
  → Dice and weighted outcomes
• The leading digit cannot be 0
  → Numbers with divisibility or digit properties

Formulas (2)

• Quadratics with random coefficients · Discriminant conditions
  $$ax^2+bx+c:\quad b^2-4ac\ \gtreqless\ 0$$
• Other conditions on random outcomes · Sum of two dice
  $$P(N=k)=\frac{6-|k-7|}{36},\quad k=2,\dots,12$$

Watch out for (2)

• Strict or not
  → Quadratics with random coefficients
• Integer endpoints
  → Other conditions on random outcomes

Formulas (3)

• Addition rule and conditional probability · Conditional probability
  $$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$$
• Independent events · Independence
  $$P(A\cap B)=P(A)\,P(B)$$
• Trials repeated until a success · Alternate turns
  $$P(A\text{ wins})=p_A+q_Aq_Bp_A+(q_Aq_B)^2p_A+\cdots=\frac{p_A}{1-q_Aq_B}$$

Watch out for (3)

• Condition on the complement
  → Addition rule and conditional probability
• Take the root that is a probability
  → Independent events
• Who throws first
  → Trials repeated until a success

Formulas (3)

• Total probability · Total probability
  $$P(E)=\sum_{i}P(H_i)\,P(E\mid H_i)$$
• Bayes' theorem · Bayes' theorem
  $$P(H_k\mid E)=\frac{P(H_k)\,P(E\mid H_k)}{\sum_iP(H_i)\,P(E\mid H_i)}$$
• Inferring an unknown bag or a lost card · Weights for an unknown bag
  $$P(k\mid\text{draw})=\frac{\binom kr}{\sum_j\binom jr}\quad(\text{equal priors})$$

Watch out for (3)

• The receiving bag has one more ball
  → Total probability
• Priors are not always equal
  → Bayes' theorem
• Include the impossible compositions
  → Inferring an unknown bag or a lost card

Formulas (2)

• Binomial probabilities · Binomial probability
  $$P(X=r)=\binom nr\,p^{r}q^{\,n-r}$$
• Recovering n and p from the mean and variance · Moments
  $$E(X)=np,\qquad\operatorname{Var}(X)=npq$$

Watch out for (2)

• Identify the trial and its p
  → Binomial probabilities
• The mean is the larger root
  → Recovering n and p from the mean and variance

Formulas (2)

• Distributions, mean and variance · Variance
  $$\operatorname{Var}(X)=E(X^2)-\big(E(X)\big)^2$$
• Special items in a sample without replacement · Sampling without replacement
  $$E(X)=\frac{nK}{N},\qquad\operatorname{Var}(X)=\frac{nK}{N}\cdot\frac{N-K}{N}\cdot\frac{N-n}{N-1}$$

Watch out for (2)

• Square the values, not the probabilities
  → Distributions, mean and variance
• Without replacement changes the variance
  → Special items in a sample without replacement