Use of Probability formulas

Basic Probability Formulas

Probability is a measure of the likelihood of a particular event happening, expressed in the range 0 to 1.
The probability of an event not happening is found by 1 minus the probability of the event: P’(A) = 1 - P(A).
The probability of the union of two events A and B is expressed as P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
The probability of an event A or event B happening, without overlapping, can be found by adding the probabilities of each event: P(A ∪ B) = P(A) + P(B).

Conditional Probability

Conditional probability refers to the probability of one event occurring, given that another event has occurred: **P(A B) = P(A ∩ B) / P(B)**.
There’s an important theorem for conditional probability known as Bayes’ Theorem, which translates to **P(A B) = P(B A) x P(A) / P(B)**.
Bear in mind that the outcome space might change depending on the knowledge about other events.

Independence

Two events are independent if the probability of one event has no effect on the probability of another.
The mathematical definition of two independent events, A and B, is P(A ∩ B) = P(A) x P(B).
If events are independent, the conditional probability of A given B is the same as the probability of A, i.e., **P(A B) = P(A)**.

Binomial Distribution

In scenarios where there are exactly two possible outcomes for each trial, we use the Binomial distribution.
Important in the calculation is the binomial coefficient, often expressed in the form ‘n choose k’.
It employs the general formula P(X = k) = C(n, k) * (p^k) * ((1 - p)^(n-k)), where n is the number of trials, p is the probability of success in each trial, k is the number of successful outcomes, and C(n, k) is the number of combinations of n items taken k at a time.

Poisson Distribution

The Poisson distribution is useful for estimating the number of events in a fixed interval of time or space.
The formula for the Poisson distribution is P(X = k) = e^-λ * λ^k / k!, where λ represents the mean number of occurrences and k is the actual number of occurrences.

As you explore these formulas, keep in mind the recommendations and definitions provided. Being comfortable with the mentioned formulas is key to tackling many problems within the realm of probability as part of the Further Mathematics.