Binomial Distribution

Probability

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

combinationsA 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

Cn

rnCr = n! r!(n-r)!

nCr = n! r!(n-r)!

This, too, can be solved using a graphic calculator...

Casio fx9750g-Plusto 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 (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...

Blaise Pascal1

1 11 2 1

1 3 3 11 4 6 4 1

1 5 10 10 5 11 6 15 20 15 6

1etcThe 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

0123456

etc.

11 1

1 2 11 3 3 1

1 4 6 4 11 5 10 10 5 1

1 6 15 20 15 6 1

etc

0123456

etc.

Combinations and Pascal’s Triangle

For reference, we need the triangle handy.

Now, consider the possible combinations from a population of 4 objects:4C0 = 1

4C1 = 44C2 = 64C3 = 44C4 = 1

the combinations are row 4 - the row number is the size of the population!

HOT TIP: always count the rows from

zero!

HOT TIP: always count the rows from

zero!

⎫⎬⎪⎪⎭

That’s better.

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-pn = number of trials

P(X = x) = nCxpxqn-x, for 0⩽x⩽1P(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 = 6x = 2p = ⅓

{ = 0.3292= 0.3292

= 0.3292

= 0.3292

STATF5F5F1F2

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

= 0.00278On 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.