The wrong belief that after a run of one result, the other result is "due" to happen next.
Precisely: The mistaken belief that past outcomes of independent random trials change the probability of the next outcome.
How it works
A coin or a die has no memory. Each toss or roll is independent, so its probabilities are the same every time, whatever came before. Long runs are simply part of randomness.
Examples
After six heads in a row, expecting tails next is the gambler's fallacy: tails is still 1/2.
Thinking "I have rolled three 6s, so I cannot get another" in Snakes and Ladders is the gambler's fallacy: a 6 is still 1/6.
Believing a lottery number is "due" because it has not come up for months is the gambler's fallacy.
Do not confuse: The law of large numbers is about proportions over very many trials; it does not make the next trial balance the last ones.
Used in
games of chance, decisions about risk, understanding randomness
A result that covers many cases at once, so that an older, narrower result becomes one of its special cases.
Precisely: A statement that extends a result to a wider class of objects; the original result is recovered from it by adding an extra condition.
How it works
Look at a result and ask what happens when one condition is removed or one more part is allowed. If a wider statement still holds, and putting the old condition back gives the old result, you have a generalisation.
Examples
(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca is a generalisation of (a + b)² = a² + 2ab + b²: put c = 0.
Brahmagupta's formula is a generalisation of Heron's formula: put d = 0.
The rectangle's area ab is a generalisation of the square's area a²: put b = a.
Do not confuse: A generalisation still has to be proved; checking a few cases is not enough. In everyday talk "generalising" can mean jumping to a conclusion; in maths it means a proved, wider result.
Used in
building new formulas, proofs, all of higher mathematics
The geometric mean of two positive numbers a and b is √(ab): the middle number that makes a, G, b a geometric progression.
Precisely: For positive numbers a and b, G = √(ab); for n positive numbers, the n-th root of their product. It never exceeds the arithmetic mean (a + b)/2, and equals it only when a = b.
How it works
In a GP each term is the one before times r, so the middle one satisfies G/a = b/G. Cross-multiplying gives G² = ab, so G = √(ab).
Examples
The geometric mean of 4 and 9 is √36 = 6: the GP 4, 6, 9 has ratio 1.5.
The geometric mean of 2 and 8 is 4, while their arithmetic mean is 5.
A price that grows 21% one year and 0% the next has grown at the geometric mean rate of 10% a year, since 1.1² = 1.21 × 1.
The geometric mean of 5 and 5 is 5, the same as the arithmetic mean.
Do not confuse: The arithmetic mean adds and halves; the geometric mean multiplies and takes the root. For 2 and 8 they are 5 and 4.
Used in
inserting terms of a GP, average growth rates, the AM-GM inequality
A sequence in which you multiply by the same number each time to get the next term.
Precisely: A sequence a, ar, ar², ar³, ..., with first term a and common ratio r; its nth term is tₙ = arⁿ⁻¹.
How it works
Find r by dividing any term by the one before. The nth term is the first term times r, (n − 1) times over. Plotted as points (n, tₙ), a GP curves: it grows faster and faster (or shrinks towards 0), never in a straight line.
Examples
1, 3, 9, 27, 81, ... is a geometric progression with a = 1 and r = 3.
In the geometric progression 2, 6, 18, ..., 4374 is the 8th term, since 2 × 3⁷ = 4374.
Bacteria doubling every hour from 30 make the geometric progression 30, 60, 120, 240, 480: 120 after 2 hours and 480 after 4.
A ball dropped from 80 m that rises to 60% each time reaches 80 × 0.6⁵ ≈ 6.22 m after the 5th bounce, a geometric progression.
Do not confuse: A GP multiplies by r; an AP adds d. 1, 2, 4, 8 is a GP, but 1, 2, 3, 4 is an AP.
Used in
growth of populations and money, bouncing and decay, fractals, sums of GPs (Class 11)
OpenStax: College Algebra 2e, 9.3 Geometric Sequences
grouped data
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Data sorted into classes (ranges), with a count for each class, instead of listing every value.
Precisely: Data presented as a frequency distribution over class intervals; summary measures are then estimated from class marks and frequencies.
How it works
Grouping makes a large set of values easy to read and graph, but the exact values are lost, so the mean, median and mode become estimates using the class marks and the formulas for grouped data.
Examples
30 students' marks shown as six classes with frequencies 2, 3, 7, 6, 6, 6 are grouped data.
The mean of that grouped data is estimated as 62 from the class marks.
A census table of ages in groups of 10 years is grouped data.
Do not confuse: Grouped data give estimates; the exact mean of the original marks might differ a little from 62.
Used in
large surveys, histograms and ogives, mean, median and mode formulas