Probability and Statistics Pointers

• Arithmetic progression


  A succession of terms formed by adding a fixed number from a preceding term. The fixed number is called the common difference.
• Arithmetic progression


  • (1, 5, 9, 13, 17, 21)
  • (2, -1, -4, -7, ...)
• Arithmetic extremes


  The first and last terms in an arithmetic progression
• Arithmetic means


  The terms in between the first and last terms in an arithmetic progression
• Finding the 10th term

  • a10 = 3 + (10-1)(5) = 48
• Finding the 9th term


  • a9 = 1 + (9-1)(3) = 25
• Finding the last term and sum

  • a10 = 2 + (10-1)(-3) = -25

  Sn = (10/2)[2(2) + (10-1)(-3)] = -115
• Geometric progression


  A sequence of numbers where each term is obtained by multiplying the preceding term by a fixed number, called the common ratio.
• Geometric progression


  • (5, 10, 20, 40, 80)

  (40, 20, 10, 5, 2.5)
• Geometric extremes


  The first and last terms in a geometric progression
• Geometric means


  The terms in between the first and last terms in a geometric progression
• Finding the 7th term

  • a7 = 2 * (3)^6 = 1458
• Finding 3 geometric means

  • 10, 20, 40
• Finding the first two terms and sum of 5 terms

  • a1 = 8, a2 = 4

  S5 = 8 * (1 - (1/4)^5) / (1 - 1/4) = 31
• Finding the sum of 6 terms

  • S6 = 3 * (1 - 3^6) / (1 - 3) = 1092
• Statistical experiment


  An experiment that generates a set of data, e.g. tossing a coin, rolling dice, drawing a card
• Sample space


  The set of all possible outcomes in a statistical experiment
• Sample spaces


  • Coin toss: {head, tail}

  Die roll: {1, 2, 3, 4, 5, 6}

  Card draw: {Ace, 2, 3, ..., King} x {club, diamond, heart, spade} x {red, black}
• Event


  A subset of the sample space
• Simple event


  An event that consists of only one element of the sample space
• Compound event


  A combination of one or more simple events
• Probability of getting a head

  • P(head) = 1/2
• Probability of drawing a red card

  • P(red card) = 26/52 = 1/2
• Multiplication counting rule


  If an experiment can be done in N1 ways, and for each of these ways, a second operation can be performed in N2 ways, then the total number of different ways in which both can be done together is N1 * N2.
• Forming 2-digit numbers


  • Without repetition: 4 * 3 = 12

  With repetition: 4 * 4 = 16
• Probability of sum 7 on two dice

  • P(sum 7) = 6/36 = 1/6
• Counting snack combinations


  • 3 * 5 * 4 * 3 = 180
• Mutually exclusive events


  Events that do not have anything in common, i.e. no intersection
• Snacks


  3 sandwiches, 5 drinks, 4 fruits, 3 candies
• Calculating number of possible snacks

  Multiply number of each type of snack: 3 sandwiches x 5 drinks x 4 fruits x 3 candies = 180
• Mutually exclusive events


  Events that do not have anything in common and do not have an intersection
• The probability of mutually exclusive events is the sum of the individual probabilities
• Mutually exclusive events


  • Getting a head or a tail in a single coin toss
• The probability of getting a head or a tail in a single coin toss is 1/2 + 1/2 = 1
• Independent events


  Successful events which do not affect nor depend on each other
• The probability of k independent events occurring is the product of the probabilities of the individual events
• Independent events


  • Getting a head on the first toss of a coin, and getting a tail on the second toss
• Permutation (nPr)


  The number of ways to arrange n distinct things taken r at a time
• Permutations


  • Arranging the letters of the word SENIOR, using 3 letters at a time
• Combination (nCr)


  The number of ways to choose r items from n items where order does not matter