Ch.10 The Binomial Formula

1. Ch.10 The Binomial Formula Sequence of Success and Fail Bernoulli Random Variable & Binomial Distribution 3. The Binomial Formula

2. 1 3 Bernoulli Random Variable & Binomial Distribution 2 INDEX Sequence of Successes and Failures The Binomial Formula

3. 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. ☞ theprocess of solution Find out all the possible ways Calculate the probability of each way Using Addition Rule, add all the calculated probabilities You can precede the second stage for convenience. 1. Sequence of Successes & Failures 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?

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

6. 1 3 Bernoulli Random Variable & Binomial Distribution 2 INDEX Sequence of Successes and Failures The Binomial Formula

7. Bernoulli Random Variable & Binomial Distribution Bernoulli Random Variable • 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

8. Bernoulli Random Variable & Binomial Distribution Binomial Distribution • 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 = X1 + X2 + X3 + … + Xn where X1, X2, … , Xn 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)

9. 1 3 Bernoulli Random Variable & Binomial Distribution 2 The Binomial Formula INDEX Sequence of Successes and Failures

10. 3. The Binomial Formula 이항공식 • 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.

11. 3. The Binomial Formula Example (1) 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? ☞ ☞n is not fixed in advance, the binomial formula does not apply.

12. 3. The Binomial Formula 예 제 (2) 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 ☞ False. Since the probability of getting 1 changes, the Binomial Formula does not apply.

13. 3. The Binomial Formula Example (2) 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 ☞ False. Trials are dependent, so the Binomial Formula does not apply.