General Statistics
Probability Distributions: Discrete Variables
The Binomial Distribution
Binomial Distribution

The binomial distribution is the relative frequency of a discrete random variable which has only two possible outcomes. As with all random variable, the mean or expected value and the variance can be calculated from the probability distribution.

Any random variable with only two possible outcomes is a binomial variable. Example a survey with only Yes / No response is a binomial variable.

1. Know the meaning of a binomial random variable, x and its distribution.

binomial experiment is an experiment with repeated trials yielding only two possible outcomes.
Properties of the binomial experiment:

1. It consist of performing some basic experiment n fixed number of times, where each experiment is called a trial. (sowe have n trials)

2. Each trial is identical and have two possible outcomes: success, S or failure, F. The probability of success, Pr[S] = p and the probability of failure, Pr[F] = q, q=1-p.

The values of p and q remain unchanged throughout each trial.

3. The trials are independent of one another.

The binomial random variable x, is the number of successes in n trials, where x = 0, 1, 2, 3, ... ,n

The frequency distribution of x is called a binomial distribution.

Examples of binomial random variables are:

(1) Results from coin toss (Head or Tail) (2) Quality of Shipment (Pass or Fail)

(3) Result of Survey (Yes or No) (4) Event study (Success or Failure)

2. Know how to calculate the probabilities of a binomial variable or distribution.

The determine the probability, P(x) or Pr[x] of a binomial random variable you must know:

(1) n, the number of trials

(2) p, the probability of success (q = 1 - p)

(3) x, the number of successes

The factorial: n! (n-factorial) = 

Then the probability of x successes is, P(x) and is given by the following formula:
, where x = 0, 1, 2,... n

You may find the Pr[x] Given you know n (number of trials) and p (proportion of successes) from the binomial reference table.

Most people use the binomial reference table. to find its probabilities rather than compute the probability directly from the formula.

Example: An airline wishes to know the probability that 4 airplane delays is the result of control reason. If the probability of such delay is 0.4 and they studied 8 aircraft departures, find the probability distributions of control related delays.

This is a binomial experiment since: (1) there are two possible outcomes: control related delays and non control related delays, (2) The probability of control related delays is 0.4, so Pr[Success] = p = 0.4 and q = 1-0.4 = 0.6, (3) The result of each airplane takeoff is an independent event, either there are no control related delay or up to 8 trials are control related delay.

So n = 8, p = 0.4 and x = 0, 1, 2, 3, 4, 5, 6, 7, 8

Therefore for x=4, the Pr[4] = 

Lookup  this probability from binomial probability table: when n=8, x=4 and p=0.40, binomial Pr[x=4] = 0.2322.
Probability Table:
x P(x)
0 0.0168
1 0.0896
2 0.209
3 0.2787
4 0.2322
5 0.1239
6 0.0413
7 0.0079
8 0.0007

3. Know how to calculate the mean or expected value and variance of a binomial distribution.

The expected value, E(x) or mean of a binomial distribution is the product of the number of trials, n and the proportion of success, p.

The variance of a binomial random variable is:

The standard deviation, 

Example. Find the expected value, the variance and standard deviation of tossing a fair coin 200 times.

This is a binomial experiment since: (1) there are two possible outcome, Head or Tail, (2) the probability of success, getting Head is P(H) = ½ = p, (so q = 1- ½ = ½ ), and (3) Each toss of the coin is an independent event. Consider each toss and experiment or trial.

So n = 200 (number of trials or independent events), p = 0.50= ½ , q = 1- p = 1- ½ = 0.5.



The standard deviation = 

4. Know how to compute the cumulative probability for a distribution.

The cumulative probability distribution is a distribution where the values of the column are obtained by adding all the preceding entries for the values of P[x].

The Cumulative Binomial Table:

Given number of trials, n  and probability of success, p, then Pr[=<x ]  or equal to an dless than x is given in the Cumulative Binomial Table:

Example if x= 3 and p=0.5 and n=8, then the cumulative Binomial Pr[x=3]= 0.8555.

You may find the cumulative Pr[x] (cum Pr[x]) Given you know n (number of trials) and p (proportion of successes) from the  interactive binomial reference table.

Otherwise you may add each probability prior to or up the Pr[x].

It is useful for determining probabilities > or < or =< or >=.

Example If the P(1) = 0.10, P(2) = 0.20, P(3) = 0.40, P(4) = 0.10 and P(5) = 0.25

Then the Probability of x < 4 = P[x<4] = P(1)+ P(2)+ P(3), the cumulative P[3]=0.60.
x P(x)=Pr[x] Cum P(x)
1 0.05 0.05
2 0.15 0.2
3 0.4 0.6
4 0.15 0.75
5 0.25 1
Probability Distribution

Cumulative Distribution