1. Ch. 6. Binomial Theory

2. 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

3. 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

4. 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

5. 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

6. Binomial Theory

7. Table 2 in Appendix (A3)

8. 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

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

10. 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

11. 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)

12. Example

13. 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.

14. 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

15. Mean and Standard Deviation of a Binomial Distribution

16. 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