1. Binomial Distributions

2. 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

3. Binomial Experiment 1.There are n identical trials 2. The purpose is to determine the number of successes.

4. 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

5. 4. The probability of the outcomes remains the same from trial to trial. 5. The trials are independent.

6. Bernoulli Trial An independent trial that has two possible outcomes: Success or failure

7. 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 - x n x

8. 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

9. P(1,1’,1’,1’) = 1 6 5 6 3

10. b) Find the probability that a one will appear in any of the four positions. P(1 any) = 1 6 5 6 3 4 1

11. c) P (exactly 2 ones show) 1 6 1 6 5 6 5 6 11 Not a 1Not a 1 P = 1 6 2 5 6 2 4 2

12. d) State the theoretical Probability Distribution for the number of ones showing in four rolls. P(x 1s in four trials) = 1 6 x 5 6 4 - x 4 x

13. 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)

14. 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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