Class 12 Mathematics Formulas PDF

INVERSE TRIGONOMETRIC FUNCTIONS FUNCTIONS DOMAIN RANGE (PRINCIPAL VALVE BRANCH) 𝒚𝒚 = 𝐬𝐬𝐬𝐬𝐬𝐬−𝟏𝟏 𝒙𝒙 [−𝟏𝟏, 𝟏𝟏] −𝝅𝝅 𝝅𝝅...

INVERSE TRIGONOMETRIC FUNCTIONS FUNCTIONS DOMAIN RANGE (PRINCIPAL VALVE BRANCH) 𝒚𝒚 = 𝐬𝐬𝐬𝐬𝐬𝐬−𝟏𝟏 𝒙𝒙 [−𝟏𝟏, 𝟏𝟏] −𝝅𝝅 𝝅𝝅 , 𝟐𝟐 𝟐𝟐 𝒚𝒚 = 𝐜𝐜𝐜𝐜𝐬𝐬 −𝟏𝟏 𝒙𝒙 [−𝟏𝟏, 𝟏𝟏] [0, 𝝅𝝅] 𝒚𝒚 = 𝐜𝐜𝐜𝐜𝐬𝐬𝐞𝐞𝐞𝐞 −𝟏𝟏 𝒙𝒙 𝑹𝑹 − (− 𝟏𝟏, 𝟏𝟏) −𝝅𝝅 𝝅𝝅 , − {𝟎𝟎} 𝟐𝟐 𝟐𝟐 𝒚𝒚 = 𝐬𝐬𝐞𝐞𝐞𝐞 −𝟏𝟏 𝒙𝒙 𝑹𝑹 − (− 𝟏𝟏, 𝟏𝟏) 𝝅𝝅 [0, 𝝅𝝅] − 𝟐𝟐 𝒚𝒚 = 𝐭𝐭𝐭𝐭𝐭𝐭−𝟏𝟏 𝒙𝒙 𝑹𝑹 −𝝅𝝅 𝝅𝝅 , 𝟐𝟐 𝟐𝟐 𝒚𝒚 = 𝐜𝐜𝐜𝐜𝐭𝐭 −𝟏𝟏 𝒙𝒙 𝑹𝑹 (0, 𝝅𝝅) 1. sin−1 (− 𝑥𝑥) = −sin−1 𝑥𝑥 tan−1 (− 𝑥𝑥) = −tan−1 𝑥𝑥 cosec −1 (− 𝑥𝑥) = −cosec −1 𝑥𝑥 2. cos −1 (− 𝑥𝑥) = π − cos −1 𝑥𝑥 cot −1 (− 𝑥𝑥) = π − cot −1 𝑥𝑥 sec −1 (− 𝑥𝑥) = π − sec −1 𝑥𝑥 π 3. sin−1 𝑥𝑥 + cos −1 𝑥𝑥 = 2 π tan−1 𝑥𝑥 + cot −1 𝑥𝑥 = 2 π cosec −1 𝑥𝑥 + sec −1 𝑥𝑥 = 2 𝑥𝑥±𝑦𝑦 4. tan−1 𝑥𝑥 ± tan−1 𝑦𝑦 = tan−1 1∓𝑥𝑥𝑥𝑥 5. sin 3𝜃𝜃 = 3 sin 𝜃𝜃 − 4 sin3 𝜃𝜃 cos 3𝜃𝜃 = 4 cos 3 𝜃𝜃 − 3 cos 𝜃𝜃 3 tan 𝜃𝜃−tan3 𝜃𝜃 tan 3𝜃𝜃 = 1−3 tan2 𝜃𝜃 2𝜃𝜃 6. 2 tan−1 𝜃𝜃 = sin−1 1+𝜃𝜃2 1 − 𝜃𝜃 2 2 tan−1 𝜃𝜃 = cos −1 1 + 𝜃𝜃 2 2𝜃𝜃 2 tan−1 𝜃𝜃 = tan−1 1 − 𝜃𝜃 2 7. 1 − cos 𝜃𝜃 = 2 sin2 𝜃𝜃/2 1 + cos 𝜃𝜃 = 2 cos 2 𝜃𝜃/2  MATRICES 1.𝑘𝑘A = 𝑘𝑘[𝑎𝑎𝑖𝑖𝑖𝑖 ]𝑚𝑚 × 𝑛𝑛 = [𝑘𝑘(𝑎𝑎𝑖𝑖𝑖𝑖 )]𝑚𝑚 × 𝑛𝑛 2. − 𝐴𝐴 = (− 1)𝐴𝐴 3. If 𝐴𝐴 = [𝑎𝑎𝑖𝑖𝑖𝑖 ]𝑚𝑚 × 𝑛𝑛 , then 𝐴𝐴′ or 𝐴𝐴′ =[𝑎𝑎𝑖𝑖𝑖𝑖 ]𝑛𝑛 × 𝑚𝑚 4. (i) (𝐴𝐴′)′ = 𝐴𝐴 (iii) (𝐴𝐴 + 𝐵𝐵)′ = 𝐴𝐴′ + 𝐵𝐵′ (ii) (𝑘𝑘𝑘𝑘)′ = 𝑘𝑘 𝐴𝐴′ (iv) (𝐴𝐴𝐴𝐴)′ = 𝐵𝐵′ 𝐴𝐴′ 5. A is a symmetric matrix if 𝐴𝐴′ = 𝐴𝐴 6. A is a skew symmetric matrix if 𝐴𝐴′ = − 𝐴𝐴. DETERMINANTS 𝑎𝑎11 𝑎𝑎12 𝑎𝑎11 𝑎𝑎12 1. Determinant of a matrix 𝐴𝐴 = 𝑎𝑎 𝑎𝑎22 is given by |𝐴𝐴| = 𝑎𝑎21 𝑎𝑎22 = 𝑎𝑎11 𝑎𝑎22 − 𝑎𝑎12 𝑎𝑎21 21 𝑎𝑎1 𝑏𝑏1 𝑐𝑐1 𝑎𝑎1 𝑏𝑏1 𝑐𝑐1 𝑏𝑏 𝑐𝑐2 𝑎𝑎2 𝑐𝑐2 2. Determinant of a matrix 𝐴𝐴 = 𝑎𝑎2 𝑏𝑏2 𝑐𝑐2 is given by |𝐴𝐴| = 𝑎𝑎2 𝑏𝑏2 𝑐𝑐2 = 𝑎𝑎1 2 − 𝑏𝑏1 𝑎𝑎 𝑐𝑐3 + 𝑏𝑏3 𝑐𝑐3 3 𝑎𝑎3 𝑏𝑏3 𝑐𝑐3 𝑎𝑎3 𝑏𝑏3 𝑐𝑐3 𝑎𝑎2 𝑏𝑏2 𝑐𝑐1 𝑎𝑎3 𝑎𝑎3 3. If 𝐴𝐴 = [𝑎𝑎𝑖𝑖𝑖𝑖 ]3 × 3 then |𝑘𝑘. 𝐴𝐴| = 𝑘𝑘 3 |𝐴𝐴| 1 4. 𝐴𝐴−1 = (adj A) |A| 𝑥𝑥1 𝑦𝑦1 1 1 5. Area of ∆ with vertices (𝑥𝑥1 , 𝑦𝑦1 ) (𝑥𝑥2 , 𝑦𝑦2 ) (𝑥𝑥3 , 𝑦𝑦3 ) is ∆ = 2 2 𝑥𝑥 𝑦𝑦2 1 𝑥𝑥3 𝑦𝑦3 1 CONTINUITY AND DIFFERENTIABILITY 𝑑𝑑 1 𝑑𝑑 1 1. 𝑑𝑑𝑑𝑑 (sin−1 𝑥𝑥) = 2. (cos −1 𝑥𝑥) = √1−𝑥𝑥 2 𝑑𝑑𝑑𝑑 2 √1−𝑥𝑥 𝑑𝑑 1 𝑑𝑑 −1 3. 𝑑𝑑𝑑𝑑 (tan−1 𝑥𝑥) = 1+𝑥𝑥 2 4. 𝑑𝑑𝑑𝑑 (cot −1 𝑥𝑥) = 1+𝑥𝑥 2 𝑑𝑑 1 𝑑𝑑 −1 5. 𝑑𝑑𝑑𝑑 (sec −1 𝑥𝑥) = 6. 𝑑𝑑𝑑𝑑 (cosec −1 𝑥𝑥) = 𝑥𝑥√𝑥𝑥 2 −1 𝑥𝑥√𝑥𝑥 2 −1 𝑑𝑑 𝑑𝑑 1 7. 𝑑𝑑𝑑𝑑 (e𝑥𝑥 ) = e𝑥𝑥 8. 𝑑𝑑𝑑𝑑 (log 𝑥𝑥) = 𝑥𝑥 9. (𝑓𝑓 ± 𝑔𝑔)(𝑥𝑥) = 𝑓𝑓(𝑥𝑥) ± 𝑔𝑔(𝑥𝑥) is continuous 10. (𝑓𝑓. 𝑔𝑔)(𝑥𝑥) = 𝑓𝑓(𝑥𝑥). 𝑔𝑔(𝑥𝑥) is continuous 𝑓𝑓 𝑓𝑓(𝑥𝑥) 11. (𝑥𝑥) = (where 𝑔𝑔(𝑥𝑥) ≠ 0) is continuous 𝑔𝑔 𝑔𝑔(𝑥𝑥)  APPLICATION OF DERIVATIVES 1. A function is said to be: (a) increasing on an interval (a, b) if 𝑥𝑥1 < 𝑥𝑥2 in (𝑎𝑎, 𝑏𝑏) ⇒ f (𝑥𝑥1 ) ≤ f (𝑥𝑥2 ) for all 𝑥𝑥1 , 𝑥𝑥2 ∈ (𝑎𝑎, 𝑏𝑏) (b) decreasing on (𝑎𝑎, 𝑏𝑏) if 𝑥𝑥1 < 𝑥𝑥2 in (𝑎𝑎, 𝑏𝑏) ⇒ f (𝑥𝑥1 ) ≥ f (𝑥𝑥2 ) for all 𝑥𝑥1 , 𝑥𝑥2 ∈ (𝑎𝑎, 𝑏𝑏) 𝑑𝑑𝑑𝑑 (𝑥𝑥−𝑥𝑥0 ) 2. The equation of the tangent vat (𝑥𝑥0 , 𝑦𝑦0 ) to the curve 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) is given by 𝑦𝑦 − 𝑦𝑦0 = 𝑑𝑑𝑑𝑑 𝑥𝑥0 𝑦𝑦0 INTEGRALS 𝑥𝑥 𝑛𝑛+1 1. ∫ 𝑥𝑥 𝑛𝑛 𝑑𝑑𝑑𝑑 = + 𝐶𝐶 𝑛𝑛 ≠ −1 2. ∫ cos 𝑥𝑥 𝑑𝑑𝑑𝑑 = sin 𝑥𝑥 + 𝐶𝐶 𝑛𝑛+1 3. ∫ sin 𝑥𝑥 𝑑𝑑𝑑𝑑 = − cos 𝑥𝑥 + 𝐶𝐶 4. ∫ sec 2 𝑥𝑥 𝑑𝑑𝑑𝑑 = tan 𝑥𝑥 + 𝐶𝐶 5. ∫ cosec −2 𝑥𝑥 𝑑𝑑𝑑𝑑 = − cot 𝑥𝑥 + 𝐶𝐶 6. ∫ sec 𝑥𝑥 tan 𝑥𝑥 𝑑𝑑𝑑𝑑 = sec 𝑥𝑥 + 𝐶𝐶 𝑑𝑑𝑑𝑑 7. ∫ cosec 𝑥𝑥 cot 𝑥𝑥 𝑑𝑑𝑑𝑑 = −cosec 𝑥𝑥 + 𝐶𝐶 8. ∫ = sin−1 𝑥𝑥 + 𝐶𝐶 √1−𝑥𝑥 2 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 9. ∫ = −cos−1 𝑥𝑥 + 𝐶𝐶 10. ∫ = tan−1 𝑥𝑥 + 𝐶𝐶 √1−𝑥𝑥 2 √1+𝑥𝑥 2 𝑑𝑑𝑑𝑑 11. ∫ = cot −1 𝑥𝑥 + 𝐶𝐶 12. ∫ 𝑒𝑒 𝑥𝑥 𝑑𝑑𝑑𝑑 = 𝑒𝑒 𝑥𝑥 + 𝐶𝐶 √1+𝑥𝑥 2 𝑎𝑎𝑥𝑥 𝑥𝑥 𝑑𝑑𝑑𝑑 13. ∫ 𝑎𝑎 𝑥𝑥 𝑑𝑑𝑑𝑑 = log 𝑎𝑎 + 𝐶𝐶 14. ∫ = sec −1 𝑥𝑥 + 𝐶𝐶 𝑥𝑥√𝑥𝑥 2 −1 𝑑𝑑𝑑𝑑 1 15. ∫ = cosec −1 𝑥𝑥 + 𝐶𝐶 16. ∫ 𝑥𝑥 𝑑𝑑𝑑𝑑 = log|𝑥𝑥| + 𝐶𝐶 𝑥𝑥√𝑥𝑥 2 −1 17. ∫ tan 𝑥𝑥 𝑑𝑑𝑑𝑑 = log|sec 𝑥𝑥| + 𝐶𝐶 18. ∫ cot 𝑥𝑥 𝑑𝑑𝑑𝑑 = log|sin 𝑥𝑥| + 𝐶𝐶 19. ∫ sec 𝑥𝑥 𝑑𝑑𝑑𝑑 = log|sec 𝑥𝑥 + tan 𝑥𝑥| 20. ∫ cosec 𝑥𝑥 𝑑𝑑𝑑𝑑 = log|cosec 𝑥𝑥 − cot 𝑥𝑥| 𝑑𝑑𝑑𝑑 1 𝑥𝑥−𝑎𝑎 𝑑𝑑𝑑𝑑 1 𝑎𝑎+𝑥𝑥 21. ∫ 𝑥𝑥 2 −𝑎𝑎2 = 2𝑎𝑎 log 𝑥𝑥+𝑎𝑎 + 𝐶𝐶 22. ∫ 𝑎𝑎2 −𝑥𝑥 2 = 2𝑎𝑎 log 𝑎𝑎−𝑥𝑥 + 𝐶𝐶 𝑑𝑑𝑑𝑑 1 𝑥𝑥 𝑑𝑑𝑑𝑑 𝑥𝑥 23. ∫ 𝑥𝑥 2 +𝑎𝑎2 = 𝑎𝑎 tan−1 𝑎𝑎 + 𝐶𝐶 24. ∫ = sin−1 𝑎𝑎 + 𝐶𝐶 √𝑎𝑎2 −𝑥𝑥 2 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 25. ∫ = log 𝑥𝑥 + √𝑥𝑥 2 − 𝑎𝑎2 + 𝐶𝐶 26. ∫ = log 𝑥𝑥 + √𝑥𝑥 2 + 𝑎𝑎2 + 𝐶𝐶 √𝑥𝑥 2 −𝑎𝑎2 √𝑥𝑥 2 −𝑎𝑎2 𝑑𝑑 27. ∫ 𝑓𝑓1 (𝑥𝑥). 𝑓𝑓2 (𝑥𝑥)𝑑𝑑𝑥𝑥 = 𝑓𝑓1 (𝑥𝑥) ∫ 𝑓𝑓2 (𝑥𝑥)𝑑𝑑𝑥𝑥 − ∫ 𝑑𝑑𝑑𝑑 𝑓𝑓1 (𝑥𝑥). ∫ 𝑓𝑓2 (𝑥𝑥)𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 28. ∫ 𝑒𝑒 𝑥𝑥 [𝑓𝑓(𝑥𝑥) + 𝑓𝑓 ′ (𝑥𝑥)]𝑑𝑑𝑥𝑥 = ∫ 𝑒𝑒 𝑥𝑥 𝑓𝑓(𝑥𝑥)𝑑𝑑𝑥𝑥 + 𝐶𝐶  𝑥𝑥 𝑎𝑎2 29. ∫ √𝑥𝑥 2 − 𝑎𝑎2 𝑑𝑑𝑑𝑑 = 𝑎𝑎 √𝑥𝑥 2 − 𝑎𝑎2 − log 𝑥𝑥 + √𝑥𝑥 2 − 𝑎𝑎2 + 𝐶𝐶 2 𝑥𝑥 𝑎𝑎2 30. ∫ √𝑥𝑥 2 + 𝑎𝑎2 𝑑𝑑𝑑𝑑 = 𝑎𝑎 √𝑥𝑥 2 + 𝑎𝑎2 + log 𝑥𝑥 + √𝑥𝑥 2 + 𝑎𝑎2 + 𝐶𝐶 2 𝑥𝑥 𝑎𝑎2 𝑥𝑥 31. ∫ √𝑎𝑎2 − 𝑥𝑥 2 𝑑𝑑𝑑𝑑 = 2 √𝑎𝑎2 − 𝑥𝑥 2 + sin−1 𝑎𝑎 + 𝐶𝐶 2 APPLICATION OF INTEGRALS 𝑏𝑏 1. Area bounded by the curve 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) , 𝑥𝑥 axis and the lines 𝑥𝑥 = 𝑎𝑎 , 𝑥𝑥 = 𝑏𝑏 (𝑏𝑏 > 𝑎𝑎) is ∫𝑎𝑎 𝑦𝑦 𝑑𝑑𝑑𝑑 = 𝑏𝑏 ∫𝑎𝑎 𝑓𝑓(𝑥𝑥) 𝑑𝑑𝑑𝑑 𝑑𝑑 𝑑𝑑 2. Area bounded by the curve 𝑥𝑥 = 𝜙𝜙(𝑦𝑦) 𝑦𝑦 axis and the lines 𝑦𝑦 = 𝑐𝑐 , 𝑦𝑦 = 𝑑𝑑 is ∫𝑐𝑐 𝑥𝑥 𝑑𝑑𝑑𝑑 = ∫𝑐𝑐 𝜙𝜙(𝑦𝑦)𝑑𝑑𝑑𝑑 3. If 𝑓𝑓(𝑥𝑥) ≥ 𝑔𝑔(𝑥𝑥) in [𝑎𝑎, 𝑐𝑐] and 𝑓𝑓(𝑥𝑥) ≤ 𝑔𝑔(𝑥𝑥) in [𝑐𝑐, 𝑏𝑏] 𝑎𝑎 < 𝑐𝑐 < 𝑏𝑏 , then area = 𝑎𝑎𝑐𝑐[𝑓𝑓(𝑥𝑥) − 𝑔𝑔(𝑥𝑥)]𝑑𝑑𝑑𝑑 + 𝑏𝑏𝑐𝑐[𝑔𝑔(𝑥𝑥) − 𝑓𝑓(𝑥𝑥)]𝑑𝑑𝑑𝑑 DIFFERENTIAL EQUATIONS 1. Order of a differential equation is the order of the highest order derivative occuring in the equation. 2. Degree of a differential equation is the highest power of the highest order derivative in it. VECTOR ALGEBRA 1. Scalar product of two vectors 𝑎𝑎⃗. 𝑏𝑏 = |𝑎𝑎⃗||𝑏𝑏 ⃗| cos 𝜃𝜃. ⃗ 2. Cross product of two vectors 𝑎𝑎⃗ × 𝑏𝑏 ⃗ =|𝑎𝑎⃗||𝑏𝑏 ⃗|sin 𝜃𝜃 𝑛𝑛 3. Position vector of A point 𝑝𝑝(𝑥𝑥, 𝑦𝑦, 𝑧𝑧) is 𝑥𝑥𝚤𝚤̂ + 𝑦𝑦𝚥𝚥̂ + 𝑧𝑧𝑘𝑘 and its magnitude by 𝑥𝑥 2 + 𝑦𝑦 2 + 𝑧𝑧 2 4. Position vector of a point R dividing the line segment PQ in the ratio 𝑚𝑚: 𝑛𝑛 ⃗ 𝑛𝑛𝑎𝑎 ⃗+𝑚𝑚𝑏𝑏 →internally, is given by 𝑚𝑚+𝑛𝑛 ⃗ −𝑛𝑛𝑎𝑎 ⃗ 𝑚𝑚𝑏𝑏 → internally, is given by 𝑚𝑚−𝑛𝑛 3D GEOMETRY 1. If 𝜃𝜃 is the actual angle 𝑟𝑟⃗ = ⃗ ⃗1 ,and 𝑟𝑟⃗ = 𝑎𝑎 𝑎𝑎1 + 𝜆𝜆 𝑏𝑏 ⃗2 then cos 𝜃𝜃 = ⃗ ⃗2 + 𝜆𝜆 𝑏𝑏 ⃗ 𝑏𝑏1.𝑏𝑏 2 ⃗ ⃗ |𝑏𝑏1 |.|𝑏𝑏2 | ⃗ ⃗ (𝑏𝑏 ×𝑏𝑏 )(𝑎𝑎 ⃗−𝑎𝑎 ⃗) ⃗1 + 𝜆𝜆 ⃗ 2. Shortest distance between 𝑟𝑟⃗ = 𝑎𝑎 ⃗2 + 𝜇𝜇 ⃗ 𝑏𝑏1 and 𝑟𝑟⃗ = 𝑎𝑎 𝑏𝑏2 is 1 ⃗2 ⃗2 1 𝑏𝑏1 ×𝑏𝑏2 𝑥𝑥−𝑥𝑥1 𝑦𝑦−𝑦𝑦 𝑧𝑧−𝑧𝑧 𝑥𝑥−𝑥𝑥 𝑦𝑦−𝑦𝑦 𝑧𝑧−𝑧𝑧2 3. Shortest distance between the lines 𝑎𝑎1 = 𝑏𝑏 1 = 𝑐𝑐 1 and 𝑎𝑎 2 = 𝑏𝑏 2 = 𝑐𝑐2 is 1 1 2 2 𝑥𝑥2 − 𝑥𝑥1 𝑦𝑦2 − 𝑦𝑦1 𝑧𝑧2 − 𝑧𝑧1 𝑎𝑎1 𝑏𝑏1 𝑐𝑐1 𝑎𝑎2 𝑏𝑏2 𝑐𝑐2 (𝑏𝑏1 𝑐𝑐2 − 𝑏𝑏2 𝑐𝑐1 )2 + (𝑐𝑐1 𝑎𝑎2 − 𝑐𝑐2 𝑎𝑎1 )2 + (𝑎𝑎1 𝑏𝑏2 − 𝑏𝑏1 𝑎𝑎2 )2 ⃗×(𝑎𝑎 𝑏𝑏 ⃗−𝑎𝑎 2 ⃗) ⃗1 + 𝜆𝜆 𝑏𝑏 ⃗ and 𝑟𝑟⃗ = ⃗ 4. Distance between parallel lines 𝑟𝑟⃗ = 𝑎𝑎 𝑎𝑎2 + 𝜇𝜇 𝑏𝑏 ⃗ is ⃗ 1 𝑏𝑏1 𝐴𝐴𝑥𝑥1 + 𝐵𝐵𝑦𝑦1 + 𝐶𝐶𝑧𝑧1 +𝐷𝐷 5. Distance from a point (𝑥𝑥1 , 𝑦𝑦1 , 𝑧𝑧1 ) to the plane 𝐴𝐴𝐴𝐴 + 𝐵𝐵𝐵𝐵 + 𝐶𝐶𝐶𝐶 + 𝐷𝐷 = 0 is √𝐴𝐴2 +𝐵𝐵2 +𝐶𝐶 2  𝑥𝑥−𝑥𝑥1 𝑦𝑦−𝑦𝑦1 𝑧𝑧−𝑧𝑧1 𝑥𝑥−𝑥𝑥2 𝑦𝑦−𝑦𝑦2 𝑧𝑧−𝑧𝑧2 6. If = = and = = are the equations of two lines then the acute angle between the 𝑙𝑙1 𝑚𝑚1 𝑛𝑛1 𝑙𝑙2 𝑚𝑚2 𝑛𝑛2 two lines is given by cos 𝜃𝜃 = |𝑙𝑙1 𝑙𝑙2 + 𝑚𝑚1 𝑚𝑚2 + 𝑛𝑛1 𝑛𝑛2 | PROBABILITY = 𝑃𝑃(𝐸𝐸 ∩ 𝐹𝐹) 1. Conditional probability of an event E, given the occurance of the event F is given by P(E|F) P(F) 2. 0 ≤ P(E|F) ≤1 𝑃𝑃(𝐸𝐸′ |𝐹𝐹) = 1 − 𝑃𝑃(𝐸𝐸|𝐹𝐹) 𝑃𝑃(𝐸𝐸 ∪ 𝐹𝐹)|𝐺𝐺) = 𝑃𝑃(𝐸𝐸)| 𝐺𝐺) + 𝑃𝑃(𝐹𝐹|𝐺𝐺) − 𝑃𝑃((𝐸𝐸 ∩ 𝐹𝐹)|𝐺𝐺) 3. P(E ∩ F)= P(E) P(F|E),P(E) ≠ 0 P(E ∩ F)= P(F) P(E|F) , P (F) ≠ 0 4. If E and F are independent, then 𝑃𝑃(𝐸𝐸 ∩ 𝐹𝐹) = 𝑃𝑃(𝐸𝐸) 𝑃𝑃(𝐹𝐹) 𝑃𝑃(𝐸𝐸|𝐹𝐹) = 𝑃𝑃(𝐸𝐸) , 𝑃𝑃(𝐹𝐹) ≠ 0 (𝐹𝐹|𝐸𝐸) = 𝑃𝑃(𝐹𝐹) , 𝑃𝑃(𝐸𝐸) ≠ 0 5. Baye's Theorem, 𝑃𝑃(𝐸𝐸𝑖𝑖 ) 𝑝𝑝(𝐴𝐴|𝐸𝐸𝑖𝑖 ) p (𝐸𝐸𝑖𝑖 |𝐴𝐴) = ∑𝑛𝑛 𝑖𝑖=1(𝐸𝐸𝑗𝑗 ) 𝑝𝑝(𝐴𝐴|𝐸𝐸𝑗𝑗 )