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One can also speak of "almost all" integers having a property to mean "all except finitely many", despite the integers not admitting a measure for which this agrees with the previous usage. For example, "almost all prime numbers are odd". There is a more complicated meaning for integers as well, discussed in the main article.
The true odds against winning for each of the three horses are 1–1, 3–2 and 9–1, respectively. In order to generate a profit on the wagers accepted, the bookmaker may decide to increase the values to 60%, 50% and 20% for the three horses, respectively. This represents the odds against each, which are 4–6, 1–1 and 4–1, in order.
The gambler's fallacy is a particular misapplication of the law of averages in which the gambler believes that a particular outcome is more likely because it has not happened recently, or (conversely) that because a particular outcome has recently occurred, it will be less likely in the immediate future.
A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable. There is not even consensus on whether mathematics is an art or a science. Some just say, "mathematics is what mathematicians do". [166] [167] A common approach is to define mathematics by its object of study. [168] [169] [170 ...
Fractional odds are written a − b (a/b or a to b), meaning a winning bettor will receive their money back plus a units for every b units they bet. Decimal odds are a single value, greater than 1, representing the amount to be paid out for each unit bet. For example, a bet of £40 at 6 − 4 (fractional odds) will pay out £40 + £60 = £100.
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Intransitive dice: One can have three dice, called A, B, and C, such that A is likely to win in a roll against B, B is likely to win in a roll against C, and C is likely to win in a roll against A. Monty Hall problem, also known as the Monty Hall paradox: [2] An unintuitive consequence of conditional probability.
Another battle often noted for being a victory against all odds was the Battle of Agincourt (1415), [10] [11] which saw a depleted English army, led by King Henry V and composed of 5,000 to 8,000 longbowmen, achieve victory over a superior French army of 15,000 to 30,000 cavalry and heavy infantry; the English were outnumbered, possibly by as ...