1. Binomial Distribution

2. Bernoulli Trials Repeated identical trials are called Bernoulli trials if: 1. There are two possible outcomes for each trial, denoted by s (for success) and f (for failure). 2. The trials are independent. 3. The probability of success remains the same from trial to trial. We denote the probability of success by the letter p. 4. The random variable can only take the values of 0 or 1.

3. A game is played by rolling a die. If the die comes up with a 1 or 2, you win the game. Otherwise you lose the game. What is the probability that you win the game? Answer: 0.333

4. To find the mean, or expectation, of a Bernoulli Distribution we use: Therefore: To find the variance of a Bernoulli Distribution we use: Therefore:

5. Mean of a Bernoulli Distribution: Variance of a Bernoulli Distribution: A Bernoulli Distribution with parameter p has a mean of p. A Bernoulli Distribution with parameter p has a variance of p(1 – p).

6. A coin is flipped three times. What are the possible combinations of getting 0 heads? 1 head? 2 heads? 3 heads? Our possibilities are: 0 heads:TTT 1 head:TTHHTTTHT 2 heads:HHTHTHTHH 3 heads:HHH A coin is flipped four times. What are the possible combinations of getting 0 heads? 1 head? 2 heads? 3 heads? 4 heads? Our possibilities are: 0 heads:TTTT 1 head:THHHHTHHHHTHHHHT 2 heads:TTHHTHTHTHHTHTTHHTHTHHTT 3 heads:THHHHTHHHHTHHHHT 4 heads:HHHH

7. Binomial Distribution The binomial distribution is the probability distribution for the number of successes in a sequence of Bernoulli trials. Suppose that n Bernoulli trials are to be performed. Then the number of outcomes with exactly x successes is equal to the binomial coefficient: Therefore, the probability of getting x successes out of n trials is:

8. Given a binomial distribution with n = 11 and p = 0.4, find the probability of getting exactly four successes. Answer: 0.236 A fair die is rolled ten times. Find the probability that exactly three 6s are scored. Answer: 0.155

9. Ninety percent of the graduates of State University who apply to a particular medical school are admitted. This year six graduates have applied for admission to the medical school. Find the probability that only four of them will be accepted. Answer: 0.0984 A library contains 2000 fiction books and 3000 non-fiction books. If eleven books are picked at random, estimate the probability that four will be fiction. Answer: 0.236

10. Mr. Galaty is taking a multiple choice exam that consists of five questions. Each question has four possible answers. He guesses at every answer. What is the probability that he passes the exam if he needs at least four correct answers to pass? Answer: 0.0156 In a school there are 600 girls and 500 boys. If ten are chosen at random to go on a trip, estimate the probability that seven of them will be girls. Answer: 0.162

11. When Mary and Jane play tennis, the probability that Mary wins a point is 0.4. Find the probability that Mary will win fewer than three of the first twelve points. Answer: 0.083 The New Progressive Party is supported by 5% of the population. Find the probability that a random sample of 30 people contains at most two supporters of the party. Answer: 0.812

12. A coin is flipped three times. What are the possible probabilities of getting a heads? xP(x) 0 1 2 3 To find the mean, or expectation, we use: Therefore: Notice that:

13. Therefore, the formula for the mean of a binomial distribution is: The variance of a discrete random variable, x, is defined by:

14. Therefore, the formula for the variance of a binomial distribution is defined as: Therefore, the formula for the standard deviation of a binomial distribution is defined as:

15. The probability that an apple, picked at random from a sack, is bad is 0.05. Find the standard deviation of the number of bad apples in a sample of 15 apples. Answer: Standard Deviation = 0.844 In a group of people the expected number who wear glasses is 2 and the variance is 1.6. Find the probability that 6 people in the group wear glasses. Answer: 0.00551

16. According to the US National Center for Health Statistic, about 60% of all eye operations are performed on females. Suppose that three patients are to be selected at random. Let x denote the number of patients out of the three chosen that are female. (a) Find the mean of the random variable x. (b) Find the standard deviation of the random variable x. Answer: Mean = 1.8 Standard Deviation = 0.849

17. When a boy plays a game at a fair, the probability that he wins a prize is 0.25. He plays the game 10 times. Let X denote the total number of prizes that he wins. Assuming that the games are independent, find: (a) E(X) (b) Answer:(a) 2.5 (b) 0.526 M03/HL1/6

18. Marian shoots ten arrows at a target. Each arrow has probability 0.4 of hitting the target, independently of all other arrows. Let X denote the number of these arrows hitting the target. (a) Find the mean and standard deviation of X. (b) Find Answer:(a) Mean = 4, Stan. Dev. = 1.55 (b) 0.954 M04/HL1/12