Binomial Distribution

Probability Binomial Distribution

topics covered... factorial calculations combinations Pascal’s Triangle Binomial Distribution tables vs calculator inverting success and failure mean and variance

factorial calculations • n!reads as “n factorial” • n! is calculated by multiplying together all Natural numbers up to and including n • For example, 6! = 1 x 2 x 3 x 4 x 5 x 6 • Factorial divisions can be simplified by cancelling equivalent factors; • eg • cancel factors • gives 10! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 6! 1 x 2 x 3 x 4 x 5 x 6 = 7 x 8 x 9 x 10 = 504

C n r combinations • A combination is a probability function. • Simply, a combination is the number of possible combinations of a specific size (r) that can be made from a set population (n). • The calculation is nCr = n! r!(n-r)! This, too, can be solved using a graphic calculator...

Casio fx9750g-Plus • to calculate Combinations, first: • RUN mode • OPTN • F6 • F3 • enter “n”, press F3, enter “r” and EXE. • For example, combination of 3 objects from a population of 5: 5C3 = 10

introducing... Blaise Pascal 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 etc 0 1 2 3 4 5 6 etc. • Blaise Pascal (1623 - 1662) was a French mathematician whose major contributions to math were in probability theory. • In physics, the SI unit for pressure is named after him, as is a high-level computer programming language. • What we will look at now is a pattern known as Pascal’s Triangle... The pattern is simple additive - each term is the sum of the two terms above it. When the rows are numbered, the first row is zero

1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 etc ⎫ ⎬ ⎪ ⎪ ⎭ 0 1 2 3 4 5 6 etc. Combinations and Pascal’s Triangle That’s better. For reference, we need the triangle handy. Now, consider the possible combinations from a population of 4 objects: 4C0 = 1 4C1 = 4 4C2 = 6 4C3 = 4 4C4 = 1 HOT TIP: always count the rows from zero! the combinations are row 4 - the row number is the size of the population!

binomial distribution Some probability distributions, for example Normal (Gaussian) distribution, describe outcome likelihoods across a range of values - continuous data. Binomial Distribution describes the range of likelihoods for events that have only two possible outcomes - success or failure. To calculate binomial probability, there must be a fixed number of independent trials, and the probability of success at each trial is constant.

So, for a Random Variable X to have an outcome value x, p = P(success) q = P(failure) = 1-p n = number of trials P(X = x) = nCxpxqn-x, for 0⩽x⩽1 Binomial probabilities can be calculated easily using either tables or calculator...

{ For single events... for example, 6C2(⅓)2(⅔)4 n = 6 x = 2 p = ⅓ STAT F5 F5 F1 = 0.3292 F2 = 0.3292

for cumulative probabilities... If using tables, either add all probabilities below the maximum value, or use the complement - add the probabilities above the value, and subtract from 1.0. This is called a success-failure inversion. For example... For a random variable with n = 12 and p = 0.75, P(X<5) = P(X=1) + P(X=2) + P(X=3) +P(X=4) = 0.000002 + 0.000035 + 0.00035405 + 0.0023898 = 0.00278 On the calculator, once in the Bin Dist menu use F2 for BCd

an example of success-failure inversion... If the random variable has 23 trials and the probability of success is 0.58, then P(X<20) is the complement of P(X>19), = 1 - P(X>19) and P(X>19) = P(X=20) + P(X=21) + P(X=22) + P(X=23) of course, you could just use the calculator, and you will not have to muck around with complements. It is just a useful technique to be aware of.

Binomial Mean and Variance avoiding the messy algebraic proofs, we have: • Mean: μ = np • Variance: σ² = npq • Standard Deviation: σ =√npq and that’s about it.

Binomial Distribution