Ch. 6. Binomial Theory

Discrete and Continuous Discrete variable- is counted Continuous variable- is measured Are the following discrete or continuous? Number of credits earned Heights of students in class Distance traveled to class tonight Number of students in class Preview: Ch. 6 –binomial –discrete Ch. 7 – normal -- continuous

Recall the 2 coin example from Ch. 4 Let X= number of heads New terminology: We call X a “Random Variable” Note: this variable is discrete P(head)= ½ for each coin P(X=0) = 1/4 P(X=1)= 2/4 P(X=2) = ¼

Recall 3 coin example Let X= number of heads P(head)= ½ for each coin P(X=0) = 1/8 P(X=1)= 3/8 P(X=2) = 3/8 P(X=3) = 1/8

Probability Distributions Are these probability distributions? Ex 1: P(X=0) = .25, P(X=1) = .6, P(X=2) = .15 Ex 2 : P(X=0) = .2, P(X=1) = .5, P(X=2) = .1 Ex 3: P(X=0) = .4, P(X=1) = - 0.2, P(X=2) = .8 Ex 4: P(X=0) = .2, P(X=1) = 0, P(X=2) = .8 Ex 5: P(X=0) = 1, P(X=1) = 0, P(X=2) = 0 Ex 6: P(X=0) = .2, P(X=1) = .7, P(X=2) = .1

Binomial Theory

Table 2 in Appendix (A3)

3 coin example- binomial theory Let X= number of heads P(head)= ½ for each coin P(X=0) = 3C0 * (1/2) 0 (1/2) 3 = 1/8 P(X=1)= 3C1 * (1/2) 1 (1/2) 2 = 3/8 P(X=2) = 3C2 * (1/2) 2 (1/2) 1 = 3/8 P(X=3) = 3C3 * (1/2) 3 (1/2) 1 = 1/8

See p=.5 column for coin problems See n= 2, 3 for 2, 3 coin problems

Find the probability of observing 3 successes in 5 trials if p = 0. 7 Find the probability of observing 3 successes in 5 trials if p = 0.7. If n=5, P(X=3)= 0.309

Example: On a 4 question multiple choice test with A,B,C,D,E, p=0 Example: On a 4 question multiple choice test with A,B,C,D,E, p=0.2, find P(X=3)

Example

Mean and St. Dev. of a Discrete Probability Distribution is the expected value of x = is the standard deviation of x = We’ll calculate the means, but see book for standard deviation examples.

Calculate the mean in the 3 coin ex = X P(X) X*P(X) .125 1 .375 3.75 2 .75 3 SUM 1.5

Mean and Standard Deviation of a Binomial Distribution

For binomial problems Mean= St Dev = Example: When tossing 6 coins, n = 6, p(head)=.5, q(tail)= .5, Mean = 6(.5)= 3 heads St Dev = = 1.22