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Binomial Distribution 1. Binomial Experiment • Definition An experiment for which following 4 conditions are satisfied is called a binomial experiment. C1. The experiment consists of “n” smaller experiments which is called trial. C2. Each trial has only two outcomes which we denote by success(S) or failure(F). C3. The trials are independent, that is the outcome on any particular trial does not influence the outcome on any other trial. C4. The probability of success is constant from trial to trial. We denote the success probability by p. • Example 1 Toss a balanced coin 10 times and count the number X of heads. Then, X = 0, 1, 2, · · · , 10. Is this a binomial experiment? (sol) We need to check the four conditions for binomial experiments. – There are total n=10 trials. – Each trial has only two outcomes, which are “HEAD” or “TAIL”. – Successive tosses are independent. – The probability of “HEAD” in each trial is p=0.5. 1 • Example 2 There are 5 white balls and 5 black balls in a box. We draw 3 balls from the box. Let X be the number of White balls among 3 drawn balls. There are two drawing methods. – Method 1(With replacement). The ball drawn from each trial is put back into the box for next draw. – Method 2(Without replacement). The ball drawn from each trial is not restored into the box for next draw. 2 2. Binomial random variable • Definition X= the number of success among n independent trials with the success probability p. X is called by Binomial Random Variable and its distribution is called by Binomial Distribution. We denote it as X ∼ B n, p and n k P X =k = p (1 − p)n−k k • Example 1 4 fair coins are tossed. What is the probability we have exactly 2 heads? • Example 2 Suppose that 20% of all copies of a partitular text book fail a certain binding strength. When 15 books are randomly selected, compute the probability that 2 among 15 books fail the test. X = the number of books which 2fail13the test among 15 selected. Hence, 15 ∼ B 15, 0.2) P X = 2 = 2 0.2 0.8 . • Example 3 75% of all customers in Papa Jones use credit card. At a certain night, the store has 20 customers. Then, what is the probability that more than 15 customers use credit card? 3 • Example 4 25% of drivers do complete stop at a certain intersection. A policeman is watching the intersection for a ticket. Suppose 20 cars pass the intersection. Then, what is the probability the policeman earns more than 10 tickets? 4 3. Formula for the Binomial Distribution If X ∼ B n, p , then • Probability Distribution Function: n k P X =k = p (1 − p)n−k k • Mean and Variance: E X =n·p V (X) = n · p · (1 − p). EX: Calculate the means and variances in previous examples. 5 4. Examples of Binomial random variable • Problem 1 25% of all drivers come to a complete stop at an intersection. 20 randomly chosen drivers. What it the probability that at most 6 will do a complete stop? What is the probability that at least 2 drivers will come to a complete stop? How many of the next 20 drivers do you expect to come to a complete stop? • Problem 2 2% of the 2 million students needs a special accomodation due to their disability. Consider a random sample of 25 students. What is the probability at least 1 received a special accommodataion? • Problem 3 20% of all telephones of a certain type are submitted for service under warranty. OF these, 60% can be repaird, whereas other other 40% should be replaced. If a company purchase 10 of these telephones, what is the probability that exactly two will end up being replaced during the wararranty? • Problem 4 An airport limo can accommodate up to four passengers. The company will accept 6 reservations. It is known that 20% of those making reservations do not appear. If the 6 researvations are made, what is the probability that at least one individual with reservation can not be accommodated on the trip? What is the expected number of available places when the limo departs. 6