Probability - Revision history

Conservative at 21:22, August 6, 2025

2025-08-06T21:22:24Z

Conservative: /* Probabilistic thinking */

2025-08-06T20:57:26Z

Probabilistic thinking

Conservative: /* Empirical vs Theoretical Probability */

2025-08-06T20:56:45Z

Empirical vs Theoretical Probability

Aschlafly at 03:58, August 30, 2020

2020-08-30T03:58:59Z

DavidB4-bot: /* See also */clean up & uniformity

2016-07-13T17:52:32Z

Aschlafly at 04:49, November 26, 2015

2015-11-26T04:49:06Z

Aschlafly: history

2015-11-26T04:48:30Z

history

Aschlafly: added good link to Table of symbols used in probability and statistics

2015-11-21T14:00:40Z

added good link to Table of symbols used in probability and statistics

GregG: Fix error; remove statement that is not mathematically correct

2013-08-26T00:34:41Z

Fix error; remove statement that is not mathematically correct

GregG: add kolmogorov probability

2012-11-25T03:22:03Z

add kolmogorov probability

← Older revision		Revision as of 21:22, August 6, 2025
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		+	[[Image:Dnddice.jpg\|thumb\|right\|250px\|The probability of rolling any specific number on a standard six-sided die is 1/6.]]
	'''Probability''' theory is the study of [[uncertainty]], a relatively new branch of mathematics.		'''Probability''' theory is the study of [[uncertainty]], a relatively new branch of mathematics.

← Older revision		Revision as of 20:57, August 6, 2025
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	[[Probabilistic thinking]] is a way of approaching situations and potential problems by considering the likelihood of a different range of outcomes on a continuum rather than assuming a single, certain outcome. It involves acknowledging uncertainty, estimating probabilities, and making sound and wise decisions based on expected value. This approach can lead to more informed decisions and better management of [[risk]] in various aspects of life.<ref>[https://www.google.com/search?q=probabilistic+thinking&oq=probabilistic+thinking&gs_lcrp=EgZjaHJvbWUyCQgAEEUYORiABDIHCAEQABiABDIHCAIQABiABDIICAMQABgWGB4yCggEEAAYChgWGB4yCAgFEAAYFhgeMggIBhAAGBYYHjIICAcQABgWGB4yCAgIEAAYFhgeMgYICRAuGEDSAQg4NjIyajBqMagCALACAA&sourceid=chrome&ie=UTF-8 Probabilistic thinking]</ref>		[[Probabilistic thinking]] is a way of approaching situations and potential problems by considering the likelihood of a different range of outcomes on a continuum rather than assuming a single, certain outcome. It involves acknowledging uncertainty, estimating probabilities, and making sound and wise decisions based on expected value. This approach can lead to more informed decisions and better management of [[risk]] in various aspects of life.<ref>[https://www.google.com/search?q=probabilistic+thinking&oq=probabilistic+thinking&gs_lcrp=EgZjaHJvbWUyCQgAEEUYORiABDIHCAEQABiABDIHCAIQABiABDIICAMQABgWGB4yCggEEAAYChgWGB4yCAgFEAAYFhgeMggIBhAAGBYYHjIICAcQABgWGB4yCAgIEAAYFhgeMgYICRAuGEDSAQg4NjIyajBqMagCALACAA&sourceid=chrome&ie=UTF-8 Probabilistic thinking]</ref>
		+
		+	[[Rational thinking]] employs probabilistic thinking. The opposite of probabilistic thinking is binary/all-or-nothing thinking, having an [[optimism bias]] such as having a "lottery mentality" or being a [[pessimist]] who engages in [[negative thinking]] (See: [[Cognitive bias]]).

	==Theory of Evolution==		==Theory of Evolution==

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 Empirical (or experimental) probability is calculated simply as the ratio of ''observed'' favourable outcomes to total samples taken. Thus a pair of dice might be rolled a thousand times, with an observed occurrence of 26 pairs of sixes. This gives an empirical probability of 26/1000 or 2.60%. With a larger number of samples, the empirical probability would be expected to tend towards the theoretical value. If it does not, then either the experimental [[methodology]] or the theoretical calculation and its assumptions would be questioned. In the case of the dice, the most likely explanation would be that the dice in the experiment were not truly unbiased.<ref>http://regentsprep.org/Regents/math/ALGEBRA/APR5/theoProp.htm</ref>
 In situations more complex than a pair of dice, the basis of theoretical probability calculations can become increasingly contentious, and the approach tends to be useful only in the case of limited or absent observational data. For example, when the outbreak of a new strain of [[influenza]] occurs, theoretical calculations of expected mortality rates are quickly produced based on extrapolation from other flu strains. These are soon superseded by actual mortality observations.<ref>http://seedmagazine.com/content/article/overestimating_avian_flu/</ref>
 ==Theory of Evolution==

← Older revision		Revision as of 03:58, August 30, 2020
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−	'''Probability''' theory is the study of uncertainty, a relatively new branch of mathematics.	+	'''Probability''' theory is the study of [[uncertainty]], a relatively new branch of mathematics.

	The '''probability''' of an event is a number between 0 and 1 representing the likelihood of the event's occurrence. It may be expressed as a simple fraction, decimal fraction, or percentage. When there are a finite number of equally likely outcomes, the probability of an event is given by		The '''probability''' of an event is a number between 0 and 1 representing the likelihood of the event's occurrence. It may be expressed as a simple fraction, decimal fraction, or percentage. When there are a finite number of equally likely outcomes, the probability of an event is given by

← Older revision		Revision as of 17:52, July 13, 2016
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	*[http://www.rapidtables.com/math/symbols/Statistical_Symbols.htm Table of symbols used in probability and statistics]		*[http://www.rapidtables.com/math/symbols/Statistical_Symbols.htm Table of symbols used in probability and statistics]

−	[[~~category~~:Probability and Statistics]]	+	[[Category:Probability and Statistics]]

← Older revision		Revision as of 04:49, November 26, 2015
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−	'''Probability''' theory is the study of uncertainty, a relatively ~~modern field~~	+	'''Probability''' theory is the study of uncertainty, a relatively new branch of mathematics.

	The '''probability''' of an event is a number between 0 and 1 representing the likelihood of the event's occurrence. It may be expressed as a simple fraction, decimal fraction, or percentage. When there are a finite number of equally likely outcomes, the probability of an event is given by		The '''probability''' of an event is a number between 0 and 1 representing the likelihood of the event's occurrence. It may be expressed as a simple fraction, decimal fraction, or percentage. When there are a finite number of equally likely outcomes, the probability of an event is given by

@@ Line 1: / Line 1: @@
 The '''probability''' of an event is a number between 0 and 1 representing the likelihood of the event's occurrence.  It may be expressed as a simple fraction, decimal fraction, or percentage. When there are a finite number of equally likely outcomes, the probability of an event is given by
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 The famous mathematician Sir Frederick Hoyle once calculated that the chances of life occurring (i.e. the theoretical probability) by chance are 1 in 10 to the power of 50, or 1 in 100,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.
 ==References==
 {{reflist|1}}

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 *[[Statistics]]
 *[[Set theory]]
 [[category:Probability and Statistics]]

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-The '''probability''' of an event is a number between 0 (representing impossibility) and 1 (representing certainty), and may be expressed as a simple fraction, decimal fraction, or percentage. When there are a finite number of equally likely outcomes, the probability of an event is given by
+The '''probability''' of an event is a number between 0 and 1 representing the likelihood of the event's occurrence.  It may be expressed as a simple fraction, decimal fraction, or percentage. When there are a finite number of equally likely outcomes, the probability of an event is given by
 <math>
@@ Line 6: / Line 6: @@
 </math>.
-For example, the probability of a tossed fair coin coming up heads is &frac12; = 0.5 = 50% because there is one favorable outcome (heads) out of 2 possible outcomes (heads and tails). Another common example is standard deck of 52 playing cards: there are 13 cards in each suit: spades, hearts, clubs and diamonds. After the deck is shuffled thoroughly, the probability that the top card on the deck is a heart is  <math>13 / 52 = 1 / 4 = 0.20</math>, or 20%.
+For example, the probability of a tossed fair coin coming up heads is &frac12; = 0.5 = 50% because there is one favorable outcome (heads) out of 2 possible outcomes (heads and tails). Another common example is standard deck of 52 playing cards: there are 13 cards in each suit: spades, hearts, clubs and diamonds. After the deck is shuffled thoroughly, the probability that the top card on the deck is a heart is  <math>13 / 52 = 1 / 4 = 0.25</math>, or 25%.
 ==Formal definition of probability==

← Older revision		Revision as of 03:22, November 25, 2012
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	For example, the probability of a tossed fair coin coming up heads is ½ = 0.5 = 50% because there is one favorable outcome (heads) out of 2 possible outcomes (heads and tails). Another common example is standard deck of 52 playing cards: there are 13 cards in each suit: spades, hearts, clubs and diamonds. After the deck is shuffled thoroughly, the probability that the top card on the deck is a heart is <math>13 / 52 = 1 / 4 = 0.20</math>, or 20%.		For example, the probability of a tossed fair coin coming up heads is ½ = 0.5 = 50% because there is one favorable outcome (heads) out of 2 possible outcomes (heads and tails). Another common example is standard deck of 52 playing cards: there are 13 cards in each suit: spades, hearts, clubs and diamonds. After the deck is shuffled thoroughly, the probability that the top card on the deck is a heart is <math>13 / 52 = 1 / 4 = 0.20</math>, or 20%.
		+
		+	==Formal definition of probability==
		+	Let <math>S</math> be the sample space and <math>E</math> be a [[σ-algebra]] of events. A probability function <math>P\colon E \to \mathbb{R}</math> is a function satisfying the Kolmogorov axioms of probability:
		+	#<math>P(A) \geq 0</math> for all events <math>A \in E</math>
		+	#<math>P(S) = 1</math>
		+	#If <math>\{A_n\}</math> is a countably infinite pairwise disjoint family of events in <math>E</math>, then
		+	<math>P\left(\bigcup_{n=1}^\infty A_n\right) = \sum_{n=1}^\infty P(A_n)</math>

	==Empirical vs Theoretical Probability==		==Empirical vs Theoretical Probability==