1. Binomial Distributions Calculating the Probability of Success

2. Contents 1. How to identify binomial distributions. 2. How to calculate binomial probabilities. 3. When to use Normal approximations for binomial distributions. 2

3. 1. How to identify binomial distributions Identification 3

4. Binomial Distribution  Discrete random variable  Define X  S={0, 1, 2, …}  Binomial setting  X  B(n, p)  Key idea: Count success! 4

5. The Binomial Setting 1. “Success” or “Failure.” 2. Probability of success same for each trial. 3. Trials independent. 4. Fixed number of trials. 5

6. Characteristics  X  B(n, p)  Expected Value:  Variance: 6

7. 2. How to Calculate Binomial Probabilities Calculations 7

8. Probability Calculations Where: k is the desired count, n is the fixed number of trials, p is the probability of success, and (1-p) is the probability of failure. 8

9. Example What is the probability of tossing a fair coin five times and getting exactly three heads? 9

10. Check for Binomial Setting 1. Success is flipping a head; failure is flipping a tail. 2. The probability of flipping heads on a fair coin is 50% each time. 3. Each flip is independent. 4. There is a fixed number of trials. 10

11. Define Values In our example: k = 3 n = 5 p = 0.5 & (1-p) = 0.5 11

12. Calculations 12

13. More Calculations 13

14. Interpretation There is about a 31% chance of flipping a fair coin 5 times and getting exactly 3 heads. 14

15. Binomial Distribution Using similar calculations, we can find each probability: 15 X012345 P(X)0.0310.1560.313 0.1560.031

16. 3. When to use Normal approximations. Normal Approximations 16

17. Normal Approximations If n is large enough, X  B(n, p)  X  N( ,  ). Follow two “rules of thumb:” 1. np  10, & 2. N(1-p)  10 17

18. The End 18