Math: Probability & Statistics

PROBABILITY BASICS

Probability measures how likely something is to happen, expressed as

a number between 0 and 1.

  0 = impossible
  0.5 = equally likely / unlikely
  1 = certain

  FORMULA:
    P(event) = number of favourable outcomes / total number of outcomes

SIMPLE PROBABILITY

  Example: A box has 125 red, 201 blue, and 174 yellow paper clips.
  What is the probability of selecting yellow?

    Total = 125 + 201 + 174 = 500
    P(yellow) = 174 / 500 = 0.348 ≈ 0.35

COMPLEMENTARY PROBABILITY

The complement of an event is everything that ISN'T that event.

Their probabilities always add up to 1.

  P(NOT A) = 1 - P(A)

  Example: P(orange) = 0.46. What is P(green) if only orange and
  green marbles exist?
    P(green) = 1 - 0.46 = 0.54

  Example: Bag has 3 blue, 9 green, 6 red, 12 yellow marbles.
  P(NOT yellow)?
    Total = 30
    P(yellow) = 12/30 = 0.4
    P(NOT yellow) = 1 - 0.4 = 0.6

  OR directly: NOT yellow = 3+9+6 = 18, so 18/30 = 0.6

MEAN (AVERAGE)

  FORMULA:
    Mean = sum of all values / number of values

  Example: Monthly rainfall: 23.7, 74.3, 65.6, 55.7, 78.3, 55.7
    Sum = 353.3
    Number of values = 6
    Mean = 353.3 / 6 = 58.88 ≈ 58.9 mm

FINDING A TOTAL FROM THE MEAN

If you know the mean and the count, you can find the total.

  Total = mean × count

  Example: 8 tests, mean score 72.
    Total = 72 × 8 = 576

FINDING A MISSING VALUE FROM THE MEAN

  Step 1: Calculate the required total (mean × count)
  Step 2: Add up all the known values
  Step 3: Subtract to find the missing value

  Example: Heights of 7 players, mean = 128 cm.
  Known heights: 138, 132, 116, 138, 126, 130. Peter = ?

    Required total = 128 × 7 = 896
    Known sum = 138 + 132 + 116 + 138 + 126 + 130 = 780
    Peter = 896 - 780 = 116 cm

MEDIAN

The median is the MIDDLE VALUE when data is arranged in order.

  ODD number of values: the median is the exact middle number.
    Example: 3, 5, 7, 9, 11 → median = 7

  EVEN number of values: the median is the AVERAGE of the two
  middle numbers.
    Example: 6.09, 6.15, 6.28, 6.86, 7.04, 7.18, 7.18, 7.45
    8 values → middle = 4th and 5th values = 6.86 and 7.04
    Median = (6.86 + 7.04) / 2 = 6.95

  STEP BY STEP:
    1. Put all values in order from smallest to largest
    2. Count the values
    3. If odd: pick the middle one (position = (n+1)/2)
    4. If even: average the two middle ones (positions n/2 and n/2+1)

MEAN vs. MEDIAN

  Mean = the arithmetic average (add everything, divide by count)
  Median = the middle value when sorted

  They are DIFFERENT calculations and usually give DIFFERENT answers.
  The test asks about both — make sure you read which one they want.

KEY FORMULAS

  Probability: P = favourable / total
  Complement: P(NOT A) = 1 - P(A)
  Mean: sum / count
  Total from mean: mean × count
  Missing value: (mean × count) - sum of known values
  Median (even count): average of two middle values
  Median (odd count): the middle value