Binomial probability - AbsoluteAstronomy.com

Binomial probability typically deals with the probability

Probability

Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

of several successive decisions, each of which has two possible outcomes.

Definition

The probability of an event can be expressed as a binomial probability if its outcomes can be broken down into two probabilities p and q, where p and q are complementary (i.e. p + q = 1) For example, tossing a coin can be either heads or tails, each which have a (theoretical) probability of 0.5. Rolling a four on a six-sided can be expressed as the probability (1/6) of getting a 4 or the probability (5/6) of rolling something else.

Calculation

If an event has a probability, p, of happening, then the probability of it happening twice is p², and in general pⁿ for n successive trials.
If we want to know the probability of rolling a three times and getting two fours and one other number (in that specific order) it becomes:

However this is only sufficient for problems where the order is specific. If order is not important in the above example, then there are 3 ways that 2 rolls of four and 1 other could occur:

110

101

011

Where 1 represents a roll of four and 0 represents a non-four roll. Since there are 3 ways of achieving the same goal, the probability is 3 times that of before, or 6.9%. If order doesn't matter, then there are

(n combinations r) possible configurations. This yields the general equation for binomial trials:

General equation

The probability of getting exactly

successes in

trials is

where

is the probability of a success,

, or the probability of a failure, and

.

The value of

is complementary to

, that is

. The expression appears also in the binomial theorem

Binomial theorem

In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with , and the coefficient a of...

Example

On a 10 question multiple choice test, with 4 options per question, the probability of getting 5 and only 5 answers correct if the answers are guessed can be calculated like so:

Therefore if somebody guesses 10 answers on a multiple choice test with 4 options, they have about a 5.8% chance of getting 5 and only 5 correct answers. If 5 or more correct answers are needed to pass, then the probability of passing can be calculated by adding the probability of getting 5 (and only 5) answers correct, 6 (and only 6) answers correct, and so on up to 10 answers correct. The total probability of 5 or more correct answers is approximately 7.8%.

Estimation

There are various methods at estimating the binomial probability if the exponents are too large to calculate

Binomial approximation

One method is by approximating the probability to a normal distribution. The requirements are that

and

for an accurate answer. Approximation is done with the following equation:

Where

and

(the standard deviation of the binomial approximation) and z is the corresponding z-score.

Poisson probability function

Another possible method is approximating to a Poisson distribution. The requirements are that

and that np and npq are within 10% of each other. The formula is

where

Connection to binomial theorem

The equation for binomial probability is the same as the binomial theorem

Binomial theorem

equation, which can be used to calculate terms in Pascal's triangle

Pascal's triangle

In mathematics, Pascal's triangle is a triangular array of the binomial coefficients in a triangle. It is named after the French mathematician, Blaise Pascal...

and the expansion of binomial equations of the form

. So, if the binomial is expanded for n = 2, we get

Rewriting the equation in a trivial way:

If a represents heads and b tails, then the above shows all the possibilities and the number of possible combinations. That is, there is one way to get two heads (aa), two ways to get a head and a tail (2ab) and one way to get two tails (bb) This applies for any degree of n. Since the sum of the coefficients in the equation (a₀ + a₁ + a₂) is the total possibilities, and since each unique case has the same probability, the probability of getting 1 occurrence of a and 1 of b (1 head and 1 tail on a coin) is 2 out of 4, or 0.5. The sum of coefficients for any binomial is 2ⁿ.