5.3 THE BINOMIAL DISTRIBUTION. The properties of a Binomial Distribution:

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1 5.3 THE BINOMIAL DISTRIBUTION The properties of a Binomial Distribution: 1. There must be a fixed number of trials. 2. Each trial can only have two outcomes, the success or failure. 3. The outcomes of each trial must be independent of one another. 4. The probability of a success must remain the same for each trial. Examples of events that might follow a Binomial distribution include: 1. The number of heads obtained by tossing a coin 5 times. 2. The number of defectives items in a box contains 100 items. 3. The number of patients cured when treated with a new drug in a group of people. A random variable X is defined to have a Binomial distribution, denoted by,, if the probability distribution of X is given by Where; Mean, variance and standard deviation for the Binomial distribution Mean, Variance, DEPARTMENT OF MATHEMATICS Page 1

2 Exercise Binomial Distribution Answer by referring to the Binomial Distribution Table 1. A burglar alarm system has 6 fail-safe components. The probability of each failing is Find these probabilities: a. Exactly 3 will fail b. Fewer than 2 will fail c. None will fail 2. In a restaurant, a study found that 42% of all patrons smoked. If the seating capacity of the restaurant is 80 people, find the mean, variance and standard deviation of the number of smokers. About how many seats should be available for smoking customers? 3. It is known that 20% from a population of laboratory mice are infertile. a. Five mice are chosen at random from this population. Find the probability that exactly two of them are infertile b. What is the least number of mice needed to be chosen for the probability that the sample contains at least one infertile mouse to be greater than 0.99? The manager of Suria s Food Market guarantees that none of his cartons of eggs containing ten eggs will contain one bad egg. If a carton contains more tha one bad egg, he will replace the ten eggs and allow the customer to keep the original eggs. If the probability that an individual egg is bad is 0.05, what is the probability that the manager will have to replace a given carton of eggs? 5. One prominent physician claims that 70% of those with lung cancer are chain smokers. If his assertion is correct, a. find the probability that of 10 such patients recently admitted to a hospital, fewer than half are chain smokers b. find the probability that of 10 such patients recently admitted to a hospital, at least 5 of them are chain smokers If the probability that a new employee in a garbage disposal company is still working with the company after 1 year is 0.55, what is the probability that out of 10 newly hired people, a. 7 will be with the company after 1 year, b. 7 or more will still be with the company after 1 year? What is the expected number of newly hired people that will leave the company after one year? E(X)=4.5 DEPARTMENT OF MATHEMATICS Page 2

3 5.4 THE POISSON DISTRIBUTION Another type of important discrete distributions is the Poisson distribution. Like binomial distribution, Poisson distribution has a very wide application. It is used to describe random variables that count the number of occurrences in a particular time interval or space. The number of occurrences is proportional to the length of the interval. The occurrences are independent events. In short, a Poisson probability distribution for the number of occurrences in a given interval must satisfy the following 3 conditions: 1. The number of occurrences is a discrete random variable. 2. The occurrences are random. 3. The occurrences are independent. Examples of events that might follow a Poisson distribution include: 1. The number of computers sold at a shop during a given week. 2. The number of defective items in the next 500 items manufactured on a machine. 3. The number of customers entering a supermarket during a two-hour interval. 4. The number of telephone calls per day. 5. The number of patients arriving at a clinic in the first hour it is opened A random variable X is defined to have a Poisson distribution, denoted by, if the probability distribution of X is given by where : : 0 :. Mean, variance and standard deviation for the Poisson distribution Mean, Variance, DEPARTMENT OF MATHEMATICS Page 3

4 Example: 1. Assume that the average number of computers sold at a shop per week is 10. a Find the probability that 5 computers are sold in that shop per week b Find the mean and variance of the distribution,,. EXERCISE XERCISE: 1. A large number of 10ml sample are collected from a lake. The mean number of bacteria in 10ml of liquid is 5. Find the probability that a sample taken has a No bacteria b 1 bacterium c more than 3 bacteria,.,. 2. A proof-reader discovered 200 misprints in a book containing 800 pages. Assuming that the misprints occur at random, find the probability that a particular page contains a no misprints b 1 or 2 misprints c more than 2 misprints.,.,. 3. A motorcar repair workshop tows in an average of 5 damaged cars per week. Assuming that the number of damaged cars towed per week follows a Poisson distribution, find the probability that a exactly 5 cars are towed in a particular week b at least 5 cars are towed in a particular week c exactly 20 cars are towed in a 4 week period d for 4 successive week, at least 5 cars are towed in each week A boy fishes regularly in a lake in Petaling Jaya catches an average of 2.4 fish per hour. Assuming the number of fish he catches follows a Poisson distribution, find the probability that he catches a 2 or more fish in half an hour b between 4 to 6 fish inclusive in 90 minutes If the number of hornbills seen during a two hour walk along a trail in the Sarawak hills is a random variable having the Poisson distribution with mean 0.8, find the probability that during such a walk, one will see a no hornbills b one hornbills c up to two hornbills A teacher makes two typing errors per page on average. Find the probability that he/she makes four or more errors DEPARTMENT OF MATHEMATICS Page 4

5 7. If X P 5, find the following probabilities: a b c If X P 2.4, find the following probabilities: a b c There are 6 rainy days during the month of July on average. Find the probability that in a given month of July, there are a less than 4 rainy days, b 6 to 8 rainy days There are 300 misprints distributed randomly throughout a 500-page book. Find the probability that each page will have a exactly two misprints, b two or more misprints On average, 8 accidents occur in a period of an hour on the XYZ Expressway. Find the probability that less than 8 accidents occur between 4.30pm to 6.00pm Poisson Approximation to the Binomial Distribution A binomial distribution with parameters n and p can be approximated by a Poisson distribution with parameter if is large and is small. In general, n 50 and p<0.1 should satisfied both at the same time or np<5. The approximation is better for larger n and smaller p that is, 0., can be used to approximate the binomial distribution,, with DEPARTMENT OF MATHEMATICS Page 5

6 Example: 1. Assume that 60,0.05 a Determine whether Poisson distribution can be used to approximate binomial distribution. b Find < There are 500 students in a private college and 2% of them are foreign students. Find the probability of randomly selecting 4 foreign students from the college. Method 1 : Use binomial distribution Method 2: Using Poisson approximation A large batch of items is known to have 3% defective items. If a sample of 200 is taken, what is the probability that the sample will contain a no defective items, b 3 defective items, c more than 4 defective items? DEPARTMENT OF MATHEMATICS Page 6

7 EXERCISE: 1. A sample of 200 items is taken from a large batch. It is know that 0.5% of the items are defective. Using a Poisson approximation to the Binomial distribution, find the probabilities that are 0,1,2,3,4 and 5 defective items in the sample , , , , , Reports show that in a university, one in every 500 students using motorcycles has an accident per year. What is the probability that 3 or less accidents occur in a year if the university has a population of 1000 students using motorcycles It is known that 2% of the calls received by a receptionist are wrong numbers. Use a Poisson approximation to the Binomial distribution to determine the probability that among 250 calls, four will be wrong numbers The state government estimated that 1% of the populations are illegal immigrants. If 150 people are randomly taken as sample, find the probability that three will be illegal immigrants % of mass produced articles are defective. If these articles are packed into boxes containing 80 articles each, what proportion of the boxes are free of defective articles and what proportion contain 2 or more defective articles? If two boxes are taken randomly, what is probability that the total number of defective articles is 1? 72.6%, 4.1%, In a manufacturing process, 1 in every 1000 items is defective, on average. In a random sample 8000 items, what is the probability that a no items are defective? b more than 3 items are defective? c less than 7 items are defective? DEPARTMENT OF MATHEMATICS Page 7

8 PAST YEARS SEM 1,2011/ The number of emergency admission to a hospital each day follows a Poisson distribution with mean 2 i. The hospital has 4 beds for emergencies at the beginning of each day. Find the probability that this number is insufficient for that day [3] ii. Find the probability that there are exactly 3 emergency admissions on two successive days [2] 2. The average number of cars which stop at a petrol station is 36 per hour. By assuming that the number of cars which stop at the petrol station follows a Poisson distribution, find the probability that more than 3 cars stop at the petrol station in an interval of 20 minutes [5] SEM 3, 2010/ The number of goals scored by the Malaysia football team in two games follows a Poisson distribution with mean. If the probability that Malaysia s football team does not score any goal in two games is , find i the value of λ 1.5 [3] ii the probability that Malaysia football team scores less than 4 goals in two games [3] SEM 3, 2010/ Assume that the number of customers who enters an antique shop per hour has a Poisson distribution with mean 1.7 i Find the probability that at least 3 customers will enter the shop in a particular hour [3] ii Each morning, a worker of the shop needs 30 minutes to get ready after opening the shop. Find the probability that he will not be interrupted by a customer in that period of time? [3] iii Find the probability that from am to 1.00 pm on a particular day, at least 3 customers entered the shop each hour [2] 90% of the customers who enter the shop do not buy anything. If 10 customers enter the shop on a particular day, find the probability that less than 3 customers buy something [3] DEPARTMENT OF MATHEMATICS Page 8