User:AirdishStraus/Sandbox

Examples
1 \over 2

$$\textstyle 1 \over 2 $$

$$\textstyle{\frac24+\frac1{34}=\frac54 = 1\frac14}$$

$$\textstyle\frac{3}{4} + \frac{7}{15} = 1\frac{13}{60}$$


 * OM (2 winners) = (2a &minus; 1) + (2b &minus; 1) = 2(a + b &minus; 1)


 * OM (1 winner) = a &minus; 1


 * OM (3 winners) = (a + 1) &times; (b + 1) &times; (c + 1) &minus; 1 &minus; (a + b + c) + 2 &times; [(a + b &minus; 1) + (a + c &minus; 1) + (b + c &minus; 1)] = (a + 1)(b + 1)(c + 1) + 3(a + b + c) &minus; 7

or more simply as OM = ab + 3(a + b) &minus; 4
 * OM (2 winners) = (a + 1) &times; (b + 1) &minus; 1 &minus; (a + b) + 2 &times; (a + b &minus; 1) + (a &minus; 1) + (b &minus; 1) = (a + 1)(b + 1) + 2(a + b) &minus; 5


 * OM (1 winner) = 2 &times; (a &minus; 1) = 2(a &minus; 1)

= (a + 1)(b + 1)(c + 1)(d + 1) + 5(a + b + c + d) &minus; 13
 * OM (4 winners) = (a + 1) &times; (b + 1) &times; (c + 1) &times; (d + 1) &minus; 1 &minus; (a + b + c + d) + 2 &times; [(a + b &minus; 1) + (a + c &minus; 1) + (a + d &minus; 1) + (b + c &minus; 1) + (b + d &minus; 1) + (c + d &minus; 1)]


 * OM (3 winners) = (a + 1) &times; (b + 1) &times; (c + 1) &minus; 1 &minus; (a + b + c) + 2 &times; [(a + b &minus; 1) + (a + c &minus; 1) + (b + c &minus; 1)] + (a &minus; 1) + (b &minus; 1) + (c &minus; 1) = (a + 1)(b + 1)(c + 1) + 4(a + b + c) &minus; 10

or more simply as OM = ab + 4(a + b) &minus; 6
 * OM (2 winners) = (a + 1) &times; (b + 1) &minus; 1 &minus; (a + b) + 2 &times; (a + b &minus; 1) + 2 &times; [(a &minus; 1) + (b &minus; 1)] = (a + 1)(b + 1) + 3(a + b) &minus; 7


 * OM (1 winner) = 3 &times; (a &minus; 1) = 3(a &minus; 1)

Short footnotes with alphabetized full citations
A list of fully-formatted citations alphabetized by author surname will be properly included for Harvard style references. The inclusion of such a section is seen as one of the advantages of that author-date referencing method. It helps the reader to determine exactly which sources have been used.

Rather than alphabetically, footnotes are simply listed in the same order in which the references were initially tagged in the article text. However, a separate alphabetized section can also be included along with the footnotes method. Where a separate alphabetical list of fully-formatted citations is maintained, then "short footnotes" may be used. That is, without giving a fully-formatted citation in the footnotes, instead the footnotes contain only author, year and page number, so the short footnotes then list references in the same format as inline Harvard references (only without the parenthesis of course). This in effect makes "short footnotes" a hybrid method that combines aspects of the different inline citations methods.

The following example shows how it can be coded. It's essentially the same as Harvard referencing, basically replacing parenthesis with tags.

  The Sun is pretty big, &lt;ref> Miller 2005, p.23&lt;/ref> however the Moon is not so big. &lt;ref> Smith 2006, p.46&lt;/ref> The Sun is also quite hot. &lt;ref> Miller 2005, p.34&lt;/ref> &#61;= References == &lt;references/> &#61;== Citations === *   *   

Result:

The Sun is pretty big, however the Moon is not so big. The Sun is also quite hot.

References (short footnotes + full citations example)
 * 1. Miller 2005, p.23
 * 2. Smith 2006, p.46
 * 3. Miller 2005, p.34

Citations

Note how each full citation is only listed once, but can be be cross-referred to multiple times from the short footnote, for example for different page references.

For the family of full cover bets that do not include singles an adjustment to the calculation is made to leave just the doubles, trebles and accumulators. Thus, a previously described winning £10 Yankee with winners at 1-3, 5-2, 6-4 and Evens has returns calculated by:

£10 &times; [(1/3 + 2) &times; (5/2 + 2) &times; (6/4 + 2) &times; (1/1 + 2) &minus; 1 &minus; [(1/3 + 1) + (5/2 + 1) + (6/4 + 1) + (1/1 + 1)]] = £999.16

In effect, the bet has been calculated as a Lucky 15 minus the singles. Note that the total returns value of £999.16 is a penny higher than the previously calculated value as this quicker method only involves rounding the final answer, and not rounding at each individual step.

In algebraic terms the OM for the Yankee bet is given by:

OM = (a + 1)(b + 1)(c + 1)(d + 1) &minus; 1 &minus; (a + b + c + d)

In the days before software became available for use by bookmakers and those settling bets in Licensed Betting Offices (LBOs) this method was virtually de rigueur for saving time and avoiding the multiple repetitious calculations necessary in settling bets of the full cover type.

Example
3 selections with decimal odds a, b and c. Expanding (a + 1)(b + 1)(c + 1) algebraically gives abc + ab + ac + bc + a + b + c + 1. This is equivalent to the OM for a Patent (treble: abc; doubles: ab, ac and bc; singles: a, b and c) plus 1. Therefore to calculate the returns for a winning Patent it is just a case of multiplying (a + 1), (b + 1) and (c + 1) together and subtracting 1 to get the OM for the winning bet. Now just multiply by the unit stake to get the total return on the bet.

E.g. The winning Patent described earlier can be more quickly and simply evaluated by the following:

Total returns = £2 &times; [(4/6 + 2) &times; (2/1 + 2) &times; (11/4 + 2) &minus; 1] = £99.33

Ignoring any bonuses, a 50 pence each-way Lucky 63 (total stake £63) with 4 winners [2-1, 5-2, 7-2 (all 1/5 odds a place) and 6-4 (1/4 odds a place)] and a further placed horse [9-2 (1/5 odds a place)] can be relatively easily calculated as follows:

Returns (win part) = 0.50 &times; [(2/1 + 2) &times; (5/2 + 2) &times; (7/2 + 2) &times; (6/4 + 2) &minus; 1] = £172.75

or more simply as 0.50 &times; (4 &times; 4.5 &times; 5.5 &times; 3.5 &minus; 1)

Returns (place part) = 0.50 &times; [(2/5 + 2) &times; (5/10 + 2) &times; (7/10 + 2) &times; (6/16 + 2) &times; (9/10 + 2) &minus; 1] = £11.79

or more simply as 0.50 &times; (2.4 &times; 2.5 &times; 2.7 &times; 2.375 &times; 2.9 &minus; 1)

Total returns = £184.54