# Difference between revisions of "Probability"

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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== |

## Revision as of 21:22, 24 November 2012

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

.

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 , or 20%.

## Contents

## Formal definition of probability

Let be the sample space and be a σ-algebra of events. A probability function is a function satisfying the Kolmogorov axioms of probability:

- for all events
- If is a countably infinite pairwise disjoint family of events in , then

## Empirical vs Theoretical Probability

Theoretical (or estimated) probability is a calculated probability based upon a set of mathematical or scientific assumptions (including other probabilities). A simple example is the theoretical probability that the chance of rolling two sixes on a pair of unbiased six-sided dice is 1/36 (2.78%).

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.^{[1]}

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.^{[2]}

## Theory of Evolution

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.