What is the formula for probability

Answer

The formula for probability is the ratio of favorable outcomes to total possible outcomes. The basic probability formula calculates how likely an event is to occur within a defined sample space.

$P(A)=\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$
$Theformulaforprobabilityistheratiooffavorableoutcomestototalpossibleoutcomes.Thebasicprobabilityformulacalculateshowlikelyaneventistooccurwithinadefinedsamplespace.$

The value of P(A) always falls between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

What is the complementary probability formula?

The complementary probability formula calculates the probability that an event does not occur. The complement of event A, denoted as A' or Aᶜ, represents all outcomes in the sample space that are not part of A.

$P(A^{\prime })=1-P(A)$

The sum of an event's probability and its complement always equals 1.

What is the addition rule formula for probability?

The addition rule formula calculates the probability that at least one of two events occurs.

Addition rule for mutually exclusive events

Mutually exclusive events cannot occur simultaneously. The formula is:

$P(A\cup B)=P(A)+P(B)$

Addition rule for non-mutually exclusive events

Non-mutually exclusive events can occur at the same time. The formula is:

$P(A\cup B)=P(A)+P(B)-P(A\cap B)$

The term P(A ∩ B) is subtracted to avoid counting the overlap twice.

What is the multiplication rule formula for probability?

The multiplication rule formula calculates the probability that two events both occur.

Multiplication rule for independent events

Independent events have no influence on each other's outcomes. The formula is:

$P(A\cap B)=P(A)\times P(B)$

Multiplication rule for dependent events

Dependent events influence each other's probabilities. The formula is:

$P(A\cap B)=P(A)\times P(B|A)$

P(B|A) represents the probability of B occurring given that A has occurred.

What is the conditional probability formula?

The conditional probability formula calculates the probability of an event occurring given that another event has already occurred.

$P(A|B)=\frac{P(A\cap B)}{P(B)}$

This formula requires that P(B) > 0, as division by zero is undefined.

What is Bayes' theorem formula?

Bayes' theorem formula calculates the probability of an event based on prior knowledge of conditions related to that event. The formula is:

$P(A|B)=\frac{P(B|A)\times P(A)}{P(B)}$

The extended form of Bayes' theorem, used when the sample space is partitioned into multiple events, is:

$P(A_i|B)=\frac{P(B|A_i)\times P(A_i)}{\sum _{j=1}^{n}P(B|A_j)\times P(A_j)}$

This formula allows for updating probability estimates as new information becomes available.

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