Sunday 20 September 2020

Math Formulae: Probability

 Here are some Probability related formulae:


  1. Probability = 
      No. of Favourable Outcomes
    Total no. of Possible Outcomes


  2. Random Experiments: Experimental activities, where the result may not be same, when they are repeated under identical conditions. For example, tossing two coins, throwing dice and so on. It has more than one possible outcomes (or results), and it is not possible to predict the outcome in advance

  3. Sample Space: Set of all possible outcomes of a random experiment. In case of throwing a dice,
        S = {1, 2, 3, 4, 5, 6}

  4. Event: Any subset E of a sample space S is called an event. For Example, in case of throwing a dice, if E denotes event of appearance of an odd number then,
        E = {1, 3, 5}

    Types of Events:
    • Impossible Events: The empty set ∅ denotes impossible event. Like having a 7 on throwing a dice.

    • Sure Events: The whole sample space, S, is called the sure event.

    • Simple (or elementary) Event: Event E having only one sample point of a sample space.

    • Compound Event: Event E with more than one sample point of sample space.

  5. Complementary Event: Every Event E has a complementary event E' which includes all elements of sample space, excluding those in E.
        i.e.    E'  =  S - E  =  {ω: ω ∈ S and ω ∉ E}

  6. Event A or B = A ∪ B = {ω :ω ∈ A or ω ∈ B}

  7. Event A and B = A ∩ B = {ω :ω ∈ A and ω ∈ B}

  8. Event A but not B = A - B = A ∩ B' = {ω :ω ∈ A and ω ∉ B}

  9. Mutually Exclusive Events: Two events A and B are called mutually exclusive events if the occurrence of any one of them excludes the occurrence of the other event, i.e., if they can not occur simultaneously.

  10. Exhaustive Events: If A ∪ B ∪ C = S then, A, B, and C are called exhaustive events.

  11. Equally likely outcomes: All outcomes with equal probability

  12. The probability P is a real valued function whose domain is the power set of S and range is the interval [0,1] satisfying the following rules:
    • For any event E, P (E) = 0
    • P (S) = 1
    • If E and F are mutually exclusive events, then P(E ∪ F) = P(E) + P(F).


  13. P(∅) = 0

  14. P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

  15. P[A ∪ B ∪ C] = P(A) + P(B) + P(C) - P(A ∩ B) - P(B ∩ C) - P(A ∩ C) + P(A ∩ B ∩ C)

  16. P(A') = P(not A) = 1 - P(A)

  17. P(A' ∩ B') = P(A ∪ B)' = 1 - P(A ∪ B)


  18. Conditional Probability: Probability of the event E given that F has already occurred. It is denoted by P(E|F).
          P(E|F) = 
      Number of elementary events favourable to E ∩ F
    Number of elementary events which are favourable to F
       = 
      P(E ∩ F)
    P(F)
    ,   provided P(F) ≠ 0

    • P(S|F)  =  P(F|F)  =  1
    • P((A ∪ B)|F) = P(A|F) + P(B|F) - P((A ∩ B)|F),   provided P(F) ≠ 0
    • P(E'|F)  =  1  -  P(E|F)


  19. Multiplication rule of probability:
          P(E ∩ F)  =  P(E) . P(F|E)  =  P(F) . P(E|F),
          provided P(E) ≠ 0 and P(F) ≠ 0.

  20. Multiplication rule of probability for more than two events:
          P(E ∩ F ∩ G) ∩= ∩P(E) . P(F|E) . P(G|(E ∩ F))

  21. Independent Events: Two events are independent events if probability of occurrence of one of them is not affected by occurrence of the other.
    • Thus, P(F|E)  =  P(F), provided P(E) ≠ 0
      and,   P (E|F)  =  P(E), provided P (F) ≠ 0
    • Thus, P(E ∩ F) = P(E) . P (F|E) = P(E) . P(F)
    • Three events A, B and C are said to be mutually independent, if
                  P(A ∩ B)  =  P(A) . P(B)
                  P(A ∩ C)  =  P(A) . P(C)
                  P(B ∩ C)  =  P(B) . P(C)
           and, P(A ∩ B ∩ C)  =  P(A) . P(B) . P(C)

  22. Theorem of total probability: Let {E1, E2, ....., En} be a partition of a sample space and suppose that each of E1, E2, ....., En has non zero probability. Let A be any event associated with S, then
         P(A) = P(E1) . P(A|E1) + P(E2) . P(A|E2) + ..... + P(En) . P(A|En)

  23. Bayes' theorem: If E1, E2, ....., En are events which constitute a partition of sample space S, i.e. E1, E2, ....., En are pairwise disjoint and E1 ∪ E2 ∪ ..... ∪ En = S and A is any event of nonzero probability, then,
         P(Ei|A)  = 
      P(Ei)P(A|Ei)
    Σ P(Ej)P(A|Ej)
    ,      for any i= 1, 2, 3, ..., n

To, readily access formula book on your mobile, you can open tools section on Tricky Maths App.

Saturday 19 September 2020

Math Formulae: Time, Speed, and Distance

 

  1. Distance = Speed × Time

  2. Speed = 
      Distance
    Time


  3. Time = 
      Distance
    Speed


  4. 1 km/hr = 
      5
    18
     m/s       
    1 m/s = 
      18
    5
     km/h

  5. Average Speed = 
      Total Distance
    Total Time


  6. When half distance is covered with speed of x km/h and other half with y km/h, then
          Average Speed = 
      2xy
    x + y


  7. When equal distances are covered with speed of x km/h, y km/h, and z km/h, then
          Average Speed = 
      3xyz
    xy + yz + zx


  8. When two bodies are moving in same direction with speed of x km/h and y km/h, then their
          Relative Speed = (x - y) km/h,     if x > y       Relative Speed = (y - x) km/h,     if x < y

  9. When two bodies are moving in opposite direction with speed of x km/h and y km/h, then their
          Relative Speed = (x + y) km/h


  10. Boats and Streams:
    If A is speed of current,
    B is speed of Boat,
    U is downstream speed of boat, and
    V is upstream speed of boat, then

    • U = B + A

    • V = B - A,     B > A

    • Speed of boat in still water = 
        U + V
      2


    • Speed of current = 
        U - V
      2


    •   Time taken by Boat in Downstream
      Time taken by Boat in Upstream
        =   
        Upstream Speed
      Downstream Speed





  11. Circular Motion:
    • When two bodies start moving along a circular path at same time, then time taken by the bodies to meet first time = 
        Length of circular path
      Relative Speed


    • When two bodies start moving along a circular path at same time, and take t1 and t2 time to complete one round then time taken to meet at starting point  =  LCM of t1 and t2.


To, readily access formula book on your mobile, you can open tools section in Tricky Maths App.

Friday 18 September 2020

Math Formulae: Series

 

  • Arithmetic Progression (A.P)

    1. A.P = a1, a2, a3, ...., an
      where, a2 = a + d,
      (or,     a2 = a - c.d)
      where, d = common difference, and a = first term of A.P

      Thus, A.P. = a, a + d, a + 2.d, ...., a + (n - 1).d

    2. nth term of A.P: an = a + (n - 1).d
      where, a = first term of A.P.

    3. Sum of n terms of A.P: Sn = 
        n
      2
       [2.a + (n - 1).d]  = 
        n
      2
       [a + l]
      where, l = last term (a + (n - 1).d)

    4. Arithmetic Mean: (of n terms)
          A.M. = 
        (a1 + a2 + a3 + ... + an)
      n
         = 
        Sn
      n


    5. Thus, Sn = n × A.M.

    6. If, in a question, we need to assume 3 consecutive terms of A.P, it is convinient to take them as, (a - d), a, (a + d)
      Similarly, if we need to take 5 consecutive terms of A.P, it is convinient to take them as, (a - 2d), (a - d), a, (a + d), (a + 2d)

    7. If, in a question, we need to assume 4 consecutive terms of A.P, it is convinient to take them as, (a - 3d), (a - d), (a + d), (a + 3d)
      where, common difference is 2d.

    8. If a particular number X is added to each term of A.P., then resultant sequence is also in A.P.

    9. If a particular number X is Subtracted from each term of A.P., then resultant sequence is also in A.P.

    10. If a particular number X is multiplied to each term of A.P., then resultant sequence is also in A.P.

    11. If each term of A.P. is divided by X, then resultant sequence is also in A.P, provided X ≠ 0.

    12. If each term of an A.P is added to or Subtracted from corresponding term of another A.P, then resultant sequence is also in A.P, provided that two A.P's have equal number of terms.

    13. If an A.P. (a1, a2, ...., an) is having odd number of terms, then
           A.M. = 
        a1 + an
      2
         = 
        a2 + an - 1
      2
         =  ...   = ak,
      where, ak = mid-term of A.P,
      and,   k = 
        n + 1
      2


    14. If an A.P. (a1, a2, ...., an) is having even number of terms, then
           A.M. = 
        a1 + an
      2
         = 
        a2 + an - 1
      2
         =  ...   = 
        ak + ak + 1
      2
       where, k = 
        n
      2



  • Geometric Progression (G.P)

    1. G.P = a, a.r, a.r2, ...., a.r(n - 1)
      where, Common ratio: r = 
        a2
      a
         = 
        a3
      a2
         =......= 
        an
      an - 1


    2. nth term of G.P: an = a.r(n - 1)
      where, a = first term of G.P.

    3. Sum of n terms of G.P: Sn = 
        a.(r n - 1)
      (r - 1)
         , if r > 1, and
        Sn = 
        a.(1 - r n)
      (1 - r)
         , if r < 1

    4. Geometric Mean: (of n terms)
          G.M. = (a1 . a2 . a3 ........ an1/n

    5. Thus, Sn = n × A.M.

    6. If, in a question, we need to assume 3 consecutive terms of G.P, it is convinient to take them as, 
        a
      r
      ,  a,  a.r
      Similarly, if we need to take 5 consecutive terms of G.P, it is convinient to take them as, 
        a
      2
        a
      r
      ,  a,  a.r,  a.r 2

    7. If, in a question, we need to assume 4 consecutive terms of G.P, it is convinient to take them as, 
        a
      3
        a
      r
      ,  a.r,  a.r3 where, common ratio is r2.

    8. If a particular number X is multiplied to each term of G.P., then resultant sequence is also in G.P.

    9. If each term of G.P. is divided by X, then resultant sequence is also in G.P, provided X ≠ 0.

    10. If each term of a G.P is multiplied to corresponding term of another G.P, then resultant sequence is also in G.P, provided that two G.P's have equal number of terms.

    11. If a G.P. (a1, a2, ...., an) is having odd number of terms, then
           G.M. = (a1 × an)1/2 = (a2 × an - 1)1/2 = ..... = ak
      where, ak = mid-term of G.P., and,     k = 
        n + 1
      2


    12. If an G.P. (a1, a2, ...., an) is having even number of terms, then
           G.M. = (a1 × an)1/2 = (a2 × an - 1)1/2 = ..... = (ak × ak + 1)1/2
      where, k = 
        n
      2



  • Harmonic Progression:

    1. H.P. = a1, a2, a3, ....., an

      where, 
        1
      a1
        1
      a2
        1
      a3
      , ......., 
        1
      an
       are in A.P.

    2. Harmonic mean: H.P. of three terms: a, b, c is b(mid-term).
      Here, H.P. (b) = 
        2ac
      (a + c)


To, readily access formula book on your mobile, you can open tools section in Tricky Maths App.

Thursday 17 September 2020

Math Formulae: Ratio And Proportion

 Some important Ratio and Proportion formulae:


 In ratio a : b (or  a ), a (numerator) is called antecedent and b (denominator) is called consequent.
b


 2.  Qualities to be expressed as ratio should have same units (like both should be in cm or in kg, but not one in kg and other in gram).


 3.  Ratio has no units.


 4.  Value of ratio (a : b) is not changed when both a and b are multiplied (or divide) by same number.


 5.  a/b  =  a  ×  d
c/dbc


 6.  Compounded ratio is result of multiplication of two or more ratios.


 7.  Duplicate ratio of (a : b) = ( a )2.
b


 8.  Sub-Duplicate ratio of (a : b) = ( a )1/2.
b


 9.  Triplicate ratio of (a : b) = ( a )3.
b


10.  Sub-triplicate ratio of (a : b) = ( a )1/3.
b


11.  a  =  (c + a.n) , if and only if,  a  =  c
b(d + b.n)bd


12.  (a + c)  >  a , if,  a  <  c
(b + d)bbd


13.  (a + c)  <  a , if,  a  >  c
(b + d)bbd


14.  If  a  >  1,  then,  (a + x)  <  a
b(b + x)b


15.  If  a  <  1,  then,  (a + x)  >  a
b(b + x)b


16.  If  a  >  1,  then,  (a - x)  >  a
b(b - x)b


17.  If  a  <  1,  then,  (a - x)  <  a
b(b - x)b


18.  If      a : b = a1:b1
    and   b : c = b2:c1
    then,  a : b : c = (a1.b2) : (b2.b1) : (b1.c1)


19.  If      a : b = a1:b1
    and   b : c = b2:c1
    and   c : d = c2:d1
    then,  a : b : c : d = (a1.b2.c2) : (b2.b1.c2) : (b1.c1.c2) : (b1.c1.d1)


20.  Proportion is equality of two ratios, i.e. a : b :: c : d. a and d are called extremes and b and c are called means.
  Here, a is 1st proportional, b is 2nd proportional,
            c is 3rd proportional, and d is 4th proportional.


21.  Product of extremes = Product of means, i.e. a × d = c × d


22.  If a : b = b : c then a, b, c are said to be in (continued) proportion.


23.  If a : b = b : c, then b2 = a × c. Here, b is said to be mean proportional between a and cand c is called third proportional.


24.  Invertendo: 
     If,  a  =  c , then,  b  =  d
bdac


25.  Alternando: 
     If,  a  =  c , then,  a  =  b
bdcd
26.  Componendo: 
     If,  a  =  c , then,  (a + b)  =  (c + d)
bdbd


27.  Dividendo: 
     If,  a  =  c , then,  (a - b)  =  (c - d)
bdbd


28.  Componendo and Dividendo: 
     If,  a  =  c , then,  (a + b)  =  (c + d)
bd(a - b)(c - d)


29.  Direct Variation:   If X ∝ Y, then
     X = k.Y,   or,      X  =  k
Y
     Where, k is constant of proportionality
     Also,    X1  =  X2
Y1Y2


30.  Inverse Variation:
     If    X ∝  1
Y
     then,    X  =  k,   or,      X.Y  =  k
Y
     Where, k is constant of proportionality
     Also,    X1.Y1  =  X2.Y2,    or,    

X1  =  Y2
X2Y1

To, readily access formula book on your mobile, you can open tools section in Tricky Maths App.

Wednesday 16 September 2020

Math Formulae: Profit, Loss and Discount

 Some important Profit, Loss and Discount formulae:


Terms Used: CP = Cost Price,     SP = Selling Price,     , MP = Marked Price

  1. Profit = SP - CP

  2. Profit % = 
      (SP - CP)
    CP
      × 100  = 
      profit
    CP
      × 100  

  3. Loss = CP - SP

  4. Loss % = 
      (CP - SP)
    CP
      × 100  = 
      loss
    CP
      × 100

  5. SP = 
      100 + profit % × CP
    100
       or,
    SP = 
      100 - loss % × CP
    100


  6. CP = 
      100 × SP
    100 + profit %
       or,
    CP = 
      100 × SP
    100 - loss %


  7. SP = (100 + p) % of CP; where profit is p % of CP.

  8. SP = (100 - x) % of CP; where loss is x % of CP.

  9. When two articles are sold at same price, but one at profit of x % and other at loss of x % then there will be a loss, and
    loss % = 
      x ²
    100


  10. MP = CP + Markup, i.e.
    MP = CP +(% markup on CP)

  11. Discount = % discount on MP

  12. SP = MP - Discount


To, readily access formula book on your mobile, you can open tools section in Tricky Maths App.

Tuesday 15 September 2020

Math Formulae: Percentages

 Here are some important formulae related to Percentages


  1. Fraction to percentage: Multiply fraction by 100 and put % sign.

  2. Percentage to Fraction: Divide percentage by 100, reduce fraction to simplest form and remove % sign.

  3. Decimal to percentage: Move decimal point 2 places to the right and add % sign.

  4. Percentage to Decimal: Move decimal point 2 places to the left and remove % sign.

  5. Fraction-Percentage values to remember:
          1
    2
       =  50 %, 
           1
    3
       =  33.33 %, 
          1
    4
       =  25 %,

          1
    5
       =  20 %, 
           1
    6
       =  16.66 %, 
          1
    7
       =  14.28 %,

          1
    8
       =  12.25 %, 
      1
    9
       =  11.11 %, 
          1
    10
       =  10 %,

          1
    11
       =  9.09 %, 
      1
    12
       =  8.33 %, 
          1
    13
       =  7.69 %,

          1
    14
       =  7.14 %, 
      1
    15
       =  6.66 %, 
          1
    16
       =  6.25 %,

          1
    17
       =  5.88 %, 
      1
    18
       =  5.55 %, 
          1
    19
       =  5.25 %,

          1
    20
       =  5 %, 
            1
    25
       =  4 %, 
               1
    50
       =  2 %,

    Examples:
    1. 6.25 % of 320 = 
      320
    16
       =  20

    2. 20 % of 630 = 
      630
    5
       =  126

  6. Quantity X as percentage of quantity Y = 
      X
    Y
     × 100

  7. What percentage of a is b = 
      b
    a
     × 100

  8. Percentage increase or decrease in a quantity = 
      Actual increase/decrease
    Original quantity
     × 100

  9. What percentage is X more than Y = 
      (X - Y)
    Y
     × 100

  10. What percentage is X less than Y = 
      (Y - X)
    Y
     × 100


To, readily access formula book on your mobile, you can open tools section in Tricky Maths App.

Monday 14 September 2020

Math Formulae: Interest

Here are some interest related formulae that can be handy in exam time:


Terms Used: SI = Simple Interest,    CI = Compound Interest,    , A = Amount

P = Principle,    t = Time,    r = rate of interest,    I = Interest
  • A = P + I

  • Simple Interest:
    1. SI = 
        P × r × t
      100


    2. A = P + 
        P × r × t
      100
         =  P 
      ( 1 + r.t) 
      100


  • Compound Interest:
    1. CI = A - P

    2. A = P 
      ( 1 + r)t
      100
         , when interest rates are compounded yearly.

    3. A = P 
      ( 1 + r/2)2t
      100
         , when interest rates are compounded half-yearly.

    4. A = P 
      ( 1 + r/4)4t
      100
         , when interest rates are compounded quarterly.

  • For first year, CI = SI

  • For two years,
       CI - SI = P 
    (r)²
    100


  • For three years,
       CI - SI = P 
    (r)²
    100
     
    (r + 3)
    100


  • Population: Total population (P) after t years,
       when population rises, P = Pi 
    (1 +  r)t
    100

       and, when it declines, P = Pi 
    (1 -  r)t
    100

    where Pi is initial population of area, and r is rate of increase or decrease of population.

  • Depriciation: Depriciation in value of a product after t years,
        Vf = Vi 
    (1 -  r)t
    100

    where, Vf is final value of product,
    Vi is initial value, and
    r is rate of depreciation in value of product.

These formulae can be found in formula book under tools section of Tricky Maths App.