3Special distributions Many real-life situations can be modelled using statistical distributions. Examples of the types of problem that can be addressed using these distributions include:In a board game, players needs a six before they can start. What is the probability that they haven’t started after 5 tries?What proportion of the adult population have an IQ above 120?The number of accidents on a stretch of motorway averages 1 every 2 days. How likely is it that there will be no accidents in a week?12% of people are left-handed. What is the probability that a class of 30 people will have more than 6 left-handed people?
4Binomial distribution This is designed to introduce the binomial distribution. Select an appropriate number of balls (e.g. 100) to drop through the machine. At each pin, each ball randomly drops to the right with probability p and to the left with probability 1 – p. Discuss with your students the shape of the distributions produced by the balls collected at the bottom. Explore the effect of changing the value of p.
5Binomial distribution Introductory example: A spinner is divided into four equal sized sections marked 1, 2, 3, 4.If the spinner is spun 6 times, how likely is it to land on 1 on four occasions?One possible sequence would be ′ 1′.Most calculators have an nCr buttonThe number of possible sequences is(i.e. the number of ways of arranging 6 items, where 4 are of one kind and 2 are of a different kind).Each sequence has probability ×So the required probability is
6Binomial distribution A binomial distribution arises when the following conditions are met:an experiment is repeated a fixed number (n) of times (i.e., there is a fixed number of trials);the outcomes from the trials are independent of one another;each trial has two possible outcomes (referred to as success and failure);the probability of a success (p) is constant.If the above conditions are satisfied and X is the random variable for the number of successes, then X has a binomial distribution. We write: X ~ B(n , p).n and p are called parameters.
7Binomial distribution Which of these situations might reasonably be modelled by a binomial distribution?1Joan takes a multiple choice examination consisting of 40 questions. X is the number of questions answered correctly if she chooses each answer completely at random.A bag contains 6 blue and 8 green counters. James randomly picks 5 counters from the bag without replacement. X is the number of blue counters picked out.A bag contains 6 blue and 8 green counters. Jan randomly picks 5 counters from the bag, replacing each counter before picking the next. X is the number of blue counters picked out.Binomial2NotbinomialOutcomes are not independent3Binomial
8Binomial distribution Which of these situations might reasonably be modelled by a binomial distribution?1NotbinomialJon throws a dice repeatedly until he obtains a six. X is the number of throws he needs before a six arises.Judy counts the number of silver cars that pass her along a busy stretch of road. X is the number of silver cars that pass in a minute.Josh is a mid-wife. He delivers 10 babies. X is the number of babies that are girls.The number of trials is not fixed2NotbinomialThe number of trials is not fixedBinomial3
9Binomial distribution Explore the effect of changing the values of n and p. Explore the shapes of the distributions produced.
10Binomial distribution If X ~ B(n , p), thenwhere q = 1 – p.Number ofpossiblesequencesProbabilityof xsuccessesProbabilityof n – xfailuresExample: X ~ B(12, 0.4).Find a) P(X = 3) b) P(X > 1).a) (to 3 s.f.)b) P(X > 1) = 1 – P(X = 0) – P(X = 1).So P(X > 1) = (3 s.f.)
11Binomial distribution Example: The probability that a baby is born a boy is A mid-wife delivers 10 babies. Find:a) the probability that exactly 4 are male;b) the probability that at least 8 are male.a)b)