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Probability rules

The Rules of Probability

  • (A ∪ B) ∪ C = A ∪ (B ∪ C)
  • A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
  • A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
  • 0 ≤ P(A) ≤ 1
  • P(A) = 1 - P'(A)
  • $P(A) = n(A)/n(U)$, where $U$ is the sample space
  • Addition rule:

    For any two events A and B:

    $P(A∪B) = P(A) + P(B) - P(A ∩ B)$

    Product rule:

    For any two independent events A and B:

    $P(A∩B) = P(A) × P(B)$

An event is an outcome from an experiment.

An experiment is the process by which an outcome is obtained.

A random experiment is one where there is uncertainty over which event may occur.

P(A) = probability of event A

P'(A) = probability of event 'not A'

P(A|B) = probability of event A given B

A random event is one which has a theoretical probability of occurring, but in practice occurs or does not occur by a pattern of chance distribution. Experimental probability does not reflect the theoretical probability well at low numbers of trials, but as the number of trials increases, the experimental probability approaches the theoretical probability.

The sample space, $U$, is a list of all possible outcomes. For example, the sample space for tossing a coin twice is: [HH, HT, TH, TT].

The probability of the event of two heads, [HH], is: $P(A) = {n(A)}/{n(U)} = 1/4$.

The probability of the event of one head and one tail, in any order, [HT] and [TH], is: $P(A) = {n(A)}/{n(U)} = 2/4 = 0.5$.

An event is expected to occur $nP$ times, where $P$ is the probability of the event occurring, and $n$ is the number of trials. e.g. out of 12 throws of a 6-sided dice, 1 specific number (e.g. 6) is expected to come up $12 × 1/6 = 2$ times.

The relative frequency is a commonly used statistic-based method of determining empirical (experimental) probability. For example, the number of accidents is divided by the total number of workdays to determine over a long sampling period the likelihood of an accident occurring on any workday in a factory.

Union of Events

Venn diagram
Probability: event sets represented pictorially by a Venn diagram

A Venn diagram is a useful way to illustrate probability events within a sample set.

Often an event can belong to more than more group.

Addition rule:

For any two events A and B:

$P(A∪B) = P(A) + P(B) - P(A ∩ B)$

Product rule:

For any two independent events A and B:

$P(A∩B) = P(A) × P(B)$

Binomial Distribution

Binomial (or Bernoulli) experiments have these characteristics:

  • ♠ Fixed number of trials $n$
  • ♥ Independency of and identical conditions for each trial
  • ♣ Two possible outcomes for each trial: success or failure
  • ⋄ For each trial, the probability $p$ of success is constant
  • ♠ The probability of failure is $q=1-p$ and constant

Bernoulli or not Bernoulli?

1. An unbiased coin is tossed ten times, and the number of heads counted: $n = 10$, $p = q = 1/2$. This is a Bernoulli experiment because a fixed number of trials, under the same conditions, with two possible outcomes, with constant probability of success or failure in each independent trial.

2. A bag contains 4 blue balls and 4 red balls. A ball is removed, the colour noted, and the ball is replaced. This procedure is repeated 5 times: $n = 5$, $p = q = 1/2$. This is a Bernoulli experiment because a fixed number of trials, under the same conditions, with two possible outcomes, with constant probability of success or failure in each independent trial.

2. A bag contains 4 blue balls and 4 red balls. A ball is removed, the colour noted, but the ball is not replaced. This procedure is repeated 5 times. This is not a Bernoulli experiment because there is not a constant probability of success or failure in each independent trial.

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