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## The Binomial Distribution

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**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) C = # of Combinations N = # of items r = how many taken at a time L,M L,N L,P M,N M,P N,P**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) Probability of 2 heads Probability of 1head and 1 tail Or Probability of 2 tails * Think about the frequency distribution.**Flip 3 Coins: 8 possible outcomes**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 H,H,H H,H,T H,T,H T,H,H T,T,H T,H,T H,T,T T,T,T 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(.009765) = .0439425 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. # Correct Probability • 0 .001 • .010 • .044 • .117 • .205 • .246 • .205 • .117 • .044 • .010 • .001 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