EDEXCEL MECHANICS 1
Name _________________________________ ObjectiveDeadlines / Progress Single probabilitiesUse nCr formula to find the number of arrangements of some objectsUse the Binomial theorem to find single probabilitiesBinomial DistributionKnow when to use a Binomial distributionUse the Binomial distribution to find probabilities and combined probabilitiesKnow how to use your calculator Find the mean and variance of the Binomial distributionSolve problems in context using the Binomial model Firstly: n different objects can be arranged in n! waysSecondly: n objects with r of one type and (n - r) of another can be arranged in nCr=?nr?=n!r!n-r!? waysWB1Find all the possible arrangements of Three objects where one is red, one is blue and one is greenFour objects where two are red, and two are blue WB2 A child has 4 yellow and 6 black identically shaped wooden bricks.The bricks are assembled into a vertical tower one brick deep.Find the number of different patterns of the bricks which can be formedWB3A bag has three ?1 coins, two 50p coins and four 10p coins. Three coins are randomly selected. Find the probability that the value of the coins is a) ?2.50 b) ?1.60 c) 30pBinomial Distribution X?~?B(n,?p)and ???PX=x=?nxpx(1-p)n-xn called the ‘index’ p is the ‘parameterWB4Find the term in the expansion of p+q12 with p7 p+q16 with p10WB5Now imagine a biased coin with P(Heads) = p If the coin is thrown 10 times we can use the Binomial theorem to find probabilities. Find a) P(10 heads) b) P( 7 heads) c) P( 4 heads)WB6Now imagine the coin is biased so that the probability of tossing heads is 0.3 What is the probability of getting 3 heads out of 5 tosses?WB7A fair die is rolled eight times. Find the probability of rolling a) no sixes b) only 3 sixes c) four twos and 4 sixesWB8We can apply this to other problems A drug is known to be effective on 60% of patients. What is the probability that 7 out of 10 treatments work?WB09 The random variable X?~?B(10,?0.2) Find PX=0PX=2PX=8Why are b) and c) not the same answer?WB10A catfood company claims that ‘9 out of ten cats prefer our brand’ if we model their claim as a Binomial model with X?~?B(n,?0.9) then what is the probability that 3 out of ten cats in a sample prefer their brand of catfoodInterpret this result WB11 A coin is biased so that the probability of tossing heads is 0.3 What are the probabilities of the different combinations when tossing it 5 times?Binomial Distribution X?~?B(5,?0.3)xP(x)Cumulative probability USING YOUR CALCULATOR fx-991 EXX ~ B(5,0.3)We can get the table of probabilities for WB 11 as follows menumenu 774: Binomial PD4: Binomial PD 1: List1: List Input 0= 1= 2= 3= 4= 5= =N : 5P : 0.3N : 5P : 0.3 Input 5= 0.3= =X ~ B(5,0.3)We can get the table of Cumulative Probabilities for WB 11 as follows menumenu 77 1: Binomial CD1: Binomial CD 1: List1: List Input 0= 1= 2= 3= 4= 5= =N : 5P : 0.3N : 5P : 0.3 Input 5= 0.3= =WB12 The random variable X ~ B(5,0.3) Find a) P(X ≤ 3) b) P(X<8) b) P(X≥6) c) P(X>4)WB13 The random variable X?~?B(20,?0.4) Find a) P(X≤7) b) P(X<6) c) P(X≥15) WB14 The random variable X?~?B(25,?0.25) Use Cumulative probabilities to find: a) P(X≤6) b) P(X=6) c) P(X>13) d) P(6<X≤10) WB15PredictionsA spinner is designed so that the probability it lands on red is 0.3MrG uses the spinner in a class competition. Mr G wants the probability of winning a prize to be less than 5%. Each member of the class has 12 spins and the number of reds is recorded Find how many reds are needed to win a prizeWB16 Alternative distribution In the production of a car it is found that 85% are without defects.The cars are produced in batches of 50.Find the probability that there are at least 40 defect-free cars in a batchWB17Alternative distribution A coin is biased so that P(heads) = 0.8Find the probability that from 15 tosses, more than 6 are headsFind the probability that from 40 tosses, 30 or more are headsWB18Problems 27% of the Forensic Biology students at Durham University studied Further Maths.Find the probability that in a sample of 8 students, a) at least 3 studied Further Maths b) between 3 and 5 studied Further Maths WB19Only 75% of sunflower seeds from a particular source produce flowers when planted. Is Binomial theory suitable to model this situation? If so, suggest suitable variables p and nb) If 10 of the seeds are planted find P(8 or more flowers)A gardener planted one row of ten sunflower seeds each week for three weeksc) What is the probability that more than 8 flowered from every row? WB20 1% of the cars manufactured on a production line are defective. Find the probability that in a sample of 200 cars, more than 2 are defectiveA company imports 200 of these cars each month. They send back the defective cars at the end of each three month period. b) What is the probability that the company sends back no cars in one particular 3 month period ................
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