### Probability: A Very Short Introduction

John Haigh, Oxford University Press 2012

Probability - formalization of the study of the thought of uncertainty.

Classical/objective view of probability - often in chance games, each outcome is equally likely, thus the same probability. The probability of an event is the proportion of outcomes that favours it.

Relative frequency - closely matches the probability. Frequency being 3 wins, relative being out of 4 games, thus 75%.

Frequentists - those who views events to occur at a characteristic frequency.

(What are frequencies, what are degrees of belief)

Odds - used when describing probabilities, but is not the same as probability as it is a ratio. Betting involves the odds(6:1) equalling the payout of 6 for each 1 you bet. A probability of 75% which is 3/4 has the odds of three to one on. Odds against is for the bet being unlikely to occur in your favour.

Subjective/probability - non negative, less than 1, percentage between 0(impossible) and 100(certain) inclusive, derived from an individual's personal judgment.

Mutually exclusive/disjoint - events have no outcomes in common.

Addition law - for disjoint events, probability of at least one is the sum of their individual probabilities.

Pairwise disjoint - A family of sets that are mutually disjoint if every set has no common elements; the intersection is an empty set. (seems similar to disjoint, investigate independent with this.)

Independent events - have no relation to events occurring before or after it.

Conditional probability - depends on the probability of an event occurring before it.

Multiplication law - the probability of a series of events occurring is the the probability of the first, multiplied by the second, conditional if the first happens.

Overlapping events - probability is event A plus event B, minus the probability of both A and B occurring.

Experiments with equally likely outcomes are repeated often, the relative frequency of the outcome should be a close match to its probability calculated objectively.

Absolute claims of probability do not exist.

The better you are at assessing probabilities when matters are uncertain, the more likely you are to be happy with the decisions you make in life.

Subjective probabilities all for different conclusions to be made for accepting or rejecting the outcome of a probability. Ex. 1 in 5 chance for surviving an operation may be a good chance for someone living in a neighbourhood that has a high rate of mortality due to gun violence. However, for another whose wealth can offset the troubles of not getting the operation, for at least the next 20 years, may decide not to take the operation.

Odds of one to one, which is a chance of 50% is said as "the odds are evens."

Values of zero and unity(100%) are the only ones that can be conclusively proved wrong by experimental evidence.

Short runs can mislead.

The classical view of probability deals with finite lists of outcomes, there is no conclusion for infinite lists.

In cards, spade and club may be disjoint but are not independent, if one occurs, the other cannot. Whereas spade and ace are independent, but not disjoint.

You can find the change of at least one event happening by working out the chance of none, the subtracting that from 100% certain, to give the remaining chance.

chapter 3 historical sketch

**Fundamentals****The workings of probability****Historical sketch****Chance experiments****Making sense of probabilities****Games people play****Applications in science, medicine and operations research****Other applications****Curiosities and dilemmas**

**Notable People:**Frederick Mosteller, zoologist Raphael Weldon, Bruno de Finetti,

**Terms:**

Classical/objective view of probability - often in chance games, each outcome is equally likely, thus the same probability. The probability of an event is the proportion of outcomes that favours it.

Relative frequency - closely matches the probability. Frequency being 3 wins, relative being out of 4 games, thus 75%.

Frequentists - those who views events to occur at a characteristic frequency.

(What are frequencies, what are degrees of belief)

Odds - used when describing probabilities, but is not the same as probability as it is a ratio. Betting involves the odds(6:1) equalling the payout of 6 for each 1 you bet. A probability of 75% which is 3/4 has the odds of three to one on. Odds against is for the bet being unlikely to occur in your favour.

Subjective/probability - non negative, less than 1, percentage between 0(impossible) and 100(certain) inclusive, derived from an individual's personal judgment.

Mutually exclusive/disjoint - events have no outcomes in common.

Addition law - for disjoint events, probability of at least one is the sum of their individual probabilities.

Pairwise disjoint - A family of sets that are mutually disjoint if every set has no common elements; the intersection is an empty set. (seems similar to disjoint, investigate independent with this.)

Independent events - have no relation to events occurring before or after it.

Conditional probability - depends on the probability of an event occurring before it.

Multiplication law - the probability of a series of events occurring is the the probability of the first, multiplied by the second, conditional if the first happens.

Overlapping events - probability is event A plus event B, minus the probability of both A and B occurring.

**Briefs:**Experiments with equally likely outcomes are repeated often, the relative frequency of the outcome should be a close match to its probability calculated objectively.

Absolute claims of probability do not exist.

The better you are at assessing probabilities when matters are uncertain, the more likely you are to be happy with the decisions you make in life.

Subjective probabilities all for different conclusions to be made for accepting or rejecting the outcome of a probability. Ex. 1 in 5 chance for surviving an operation may be a good chance for someone living in a neighbourhood that has a high rate of mortality due to gun violence. However, for another whose wealth can offset the troubles of not getting the operation, for at least the next 20 years, may decide not to take the operation.

Odds of one to one, which is a chance of 50% is said as "the odds are evens."

Values of zero and unity(100%) are the only ones that can be conclusively proved wrong by experimental evidence.

Short runs can mislead.

The classical view of probability deals with finite lists of outcomes, there is no conclusion for infinite lists.

In cards, spade and club may be disjoint but are not independent, if one occurs, the other cannot. Whereas spade and ace are independent, but not disjoint.

You can find the change of at least one event happening by working out the chance of none, the subtracting that from 100% certain, to give the remaining chance.

chapter 3 historical sketch

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