1. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics for Economist Ch.10 The Binomial Formula 1.Sequence of Success and Fail 2.Bernoulli Random Variable & Binomial Distribution 3. The Binomial Formula

2. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 2/13 INDEX 1 Sequence of Successes and Failures Bernoulli Random Variable & Binomial Distribution 2 3 The Binomial Formula

3. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 3/13 1. Sequence of Successes & Failures Binomial Formula : application examples  When a coin is tossed 4 times, the probability of getting 1 head  When a die is rolled 10 times, the probability of getting 3 aces  The probability that the price of a stock which I bought today will increase for each 5 day  When 5 drawings are made at random with replacement draws from a box containing 1 red marbles and 9 green ones, the probability that 2 draws will be red When the outcome is divided into two parts (Success & Failure), the Probability is acquired by Binomial Formula

4. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 4/13 Example A Box contains 1 red marble and 9 green ones. Five draws are made at random with replacement. What is the probability that 2 draws will be red? Find out all the possible ways Calculate the probability of each way Using Addition Rule, add all the calculated probabilities ☞ the process of solution You can precede the second stage for convenience. 1. Sequence of Successes & Failures

5. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 5/13 # of all the possible ways The probability of each way add all the calculated probabilities. Total number of trials # of success # of failures Example 1. Sequence of Successes & Failures

6. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 6/13 INDEX 1 3 Sequence of Successes and Failures Bernoulli Random Variable & Binomial Distribution 2 The Binomial Formula

7. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 7/13 2.Bernoulli Random Variable & Binomial Distribution  Bernoulli Trial  a trial in which the outcomes are divided into 2 parts  Bernoulli Random Variable  in a Bernoulli Trial, a random variable which assigns 1 to success and 0 to failure Bernoulli Random Variable

8. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 8/13  Bernoulli Random Variable If we repeat identical Bernoulli trial n times independently, and let total number of successes be X, then X is denoted X = X 1 + X 2 + X 3 + … + X n where X 1, X 2, …, X n are random variables which assign 1 if the n-th outcome is success, and 0 to the failure  If the random variable X follows the binomial distribution, we write: X ~ B ( n, p ) Binomial Distribution 2.Bernoulli Random Variable & Binomial Distribution

9. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 9/13 INDEX 13 The Binomial Formula Sequence of Successes and Failures Bernoulli Random Variable & Binomial Distribution 2

10. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 10/13 3. The Binomial Formula  The probability that success will occur k times out of n is given by the Binomial Formula n :# of trials, k :# of successes, p : probability of success  The Binomial Formula works under the following conditions  The value of n must be fixed in advance.  The trials must be independent.  p must be the same from trial to trial. 이항공식

11. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 11/13 EX1) A die is rolled 10 times. What is the probability of getting 2 aces? EX2) A die is rolled until it first lands six. If this can be done using the binomial formula, find the probability of getting 2 aces. If not, why not? Example (1) ☞ n is not fixed in advance, the binomial formula does not apply. ☞ 3. The Binomial Formula

12. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 12/13 EX3) Ten draws are made at random with replacement from the box 1 1 2 3 4 5. However, just before the last draw is made, whatever has gone on, the ticket 5 is removed from the box. True or false : the probability of drawing two 1 ’ s is 예 제 (2) ☞ False. Since the probability of getting 1 changes, the Binomial Formula does not apply. 3. The Binomial Formula

13. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 13/13 EX4) Four draws are made at random without replacement from the box in exemple 3. True or false : the probability of drawing two 1 ’ s is Example (2) ☞ False. Trials are dependent, so the Binomial Formula does not apply. 3. The Binomial Formula