1. Ch. 6. Binomial Theory-- Questions

2. Discrete or Continuous Review– Are the following discrete or continuous variables? How do you know – Number of credits earned – Heights of students in class – Distance traveled to class tonight – Number of students in class

3. Recall the 2 coin example 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) =.4, P(X=1) =.9, P(X=2) = -.1 Ex 6: P(X=0) =.2, P(X=1) =.9, P(X=2) = -.1

6. Complement of events If P(snow today)=.2, What is the P(not snow)? How are these events related? Another ex: If P(pass)=.8, P(fail)=?

7. Binomial Theory

8. An example of a binomial table

9. 3 coin example- binomial theory Let X= number of heads P(head)= ½ for each coin P(X=0) = 3 C 0 * (1/2) 0 (1/2) 3 = 1/8 P(X=1)= 3 C 1 * (1/2) 1 (1/2) 2 = 3/8 P(X=2) = 3 C 2 * (1/2) 2 (1/2) 1 = 3/8 P(X=3) = 3 C 3 * (1/2) 3 (1/2) 0 = 1/8

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

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

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

13. Mean and St. Dev. of a Discrete Probability Distribution is the expected value of x = is the standard deviation of x = See book for some general examples. We will just concentrate on a special case: the binomial theory…

14. Mean and Standard Deviation of a Binomial Distribution

15. 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

16. Mean and St Dev Example Calculate the standard deviation of a binomial population with n = 100 and p = 0.3. a). 21b).9c). 4.5825d). 4.41 Answer: C

17. Birthday problem Let E=probability that at least 2 of us have the same birthday. E complement= ?? Recall: P(E)=1-P(E complement)

18. Answer to Bday problem If n=5, P(E complement)= ___ So P(E)= ___

19. For larger n Number of people P(E-complement)- none have same birthday P(E)- At least 2 have same bday 597.3%2.7% 1088.3%11.7% 1574.7%25.3% 2058.9%41.1% 2543.1%56.9% 3029.4%70.6% 3518.6%81.4% 4010.9%89.1% 455.9%94.1% 503.0%97.0% 551.4%98.6% 3660.0%100.0%