Binomial Distributions

Quality Control engineers use the concepts of binomial testing

extensively in their examinations.

An item, when tested, has only 2 possible states: PASS or FAIL

Binomial Experiment

1. There are n identical trials

2. The purpose is to determine the number of successes.

3. There are 2 possible outcomes:

Success (p), or failure (q).

The probability of success is denoted p, and the probability of failure is q or 1 - p

4. The probability of the outcomes remains the same from trial to trial.

5. The trials are independent.

Bernoulli Trial

An independent trial that has two possible outcomes:

Success or failure

Binomial Probability Distribution

Consider a binomial experiment in which there are n Bernoulli trials, each with a probability of success of p. The probability of x successes in the n trials is given by

P(x) = (p)x(1 – p)n - xnx

Consider rolling a die 4 times.

a) What is the probability that the first roll will be a one, and all other rolls will be something other than a one?

1

6

5

6

5

6

5

6

P(1,1’,1’,1’) =1

6

5

6

3

b) Find the probability that a one will appear in any of the four positions.

P(1 any) =1

6

5

6

34

1

c) P (exactly 2 ones show)

1

6

1

6

5

6

5

6

1 1 Not a 1 Not a 1

P = 1

6

25

6

24

2

d) State the theoretical Probability Distribution for the number of ones showing in four rolls.

P(x 1s in four trials) =

1

6

x5

6

4 - x4

x

Expected Value of a Binomial Experiment

Consists of n Bernoulli Trials with a probability of success, p, on each trial is

E(X) = np

Number of trials X P(success)

Flip a coin 4 times, how many times do you expect

tails to show up?2

E(2T) = np= 4 X ½= 2

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