The Binomial Distribution Permutations: How many different pairs of two items are possible from these four letters: L, M. N, P. L,M L,N L,P M,L M,N M,P N,L N,M N,P P,L P,M P,N P = # of Permutations N = # of items r = how many taken at a time

Combinations (order is not important) L,M L,N L,P M,N M,P N,P C = # of Combinations N = # of items r = how many taken at a time

If we flip a single coin, there are two possible outcomes: head or tail (H or T). P(one head) =.5 P(no head) =.5 Together the probabilities of these possible outcomes sum to 1.0. If we flip two coins there are four possible outcomes. H,H H,T T,H T,T P(two heads) =.25 P(one head) =.50 P(no heads) =.25 Total = 1.0 Can be calculated by… P = P(H) and Q = P(T) Or Probability of 2 heads Probability of 1head and 1 tail Probability of 2 tails * Think about the frequency distribution.

Flip 3 Coins: 8 possible outcomes H,H,H H,H,T H,T,H T,H,H T,T,H T,H,T H,T,T T,T,T P(3 heads) = 1/8 or.13 P(2 heads) = 3/8 or.37 P(1 head) = 3/8 or.37 P(0 heads) = 1/8 or.37 Total = 1.0 Can be calculated by….. or

Binomial Distribution A binomial distribution is produced when each of a number of Independent trials results in one of two Mutually Exclusive outcomes. A single flip of a coin is a Bernoulli trial. They reflect a discrete variable. P(X) = Probability of X number of an outcome N = Number of trials P = Probability of success on any given trial Q = (1 – P) The number of combinations of N things taken X at a time

Example: Guess if I am thinking of the colour black or white. Because I randomly choose one of the two colours, the probability that I am thinking of black is.5 on any given trial. There will be 10 trials. What is the probability of guessing 8 of them correctly? = 45( ) = This is the probability of * OR MORE correct, but of exactly 8 out of 10 correct.

Binomial Distribution & Testing a Hypothesis Let us repeat the previous example, but now the question is what is the probability of guessing 8 or more of the trials correctly. We know that P(8) =.44 That is less than.05, But that is P(8) against 10, including P(9) and P(10). We need to test the probability of 8 or more correct. P(8) =.044 P(9) =.010 P(10) =.001 Total P =.055 The probability of guessing 8 or more correctly out of 10 turns out to have a probability greater than.05, thus we fail to reject the null hypothesis. Which was what?

Binomial Distribution We can calculate the probability of 0 to 10 correct. # CorrectProbability Mathematical Sampling Distribution

As N gets large, all binomial distributions approach normality, regardless of P. Thus, Mean = NP For example, 10(.5) = 5 That is, if you examine the data on the previous slide, you will see that the mean of the distribution is 5 (the number of correct trials). Variance = NPQ For example, 10(.5)(.05) = 2.25 Standard Deviation = = 1.58