Binomial Distribution: Past Paper Questions

this video looks at some past paper questions on the binomial distribution the first question I'm going to look at comes from the January 2008 paper which is on the exciting topic of nuts and bolts the question tells us that the probability of a bolt being faulty is not 0.3 and that we're looking at a random sample of 20 volts the first part of the question asks us to calculate the probability that exactly 2 out of these 20 bolts are faulty well the first thing is to create a random variable X which is equal to the number of faulty bolts because then we can say that X has the binomial distribution with 20 trials and not 0.3 as the probability of success it's 20 trials because there are 20 bolts and not 0.3 because that's the probability of a bolt being faulty now the question is asking us for the probability that there are exactly two faulty bolts and that's the probability that X is equal to 2 we know from the formula that that's equal to 20 choose 2 times not 0.3 squared x not 0.7 to the power of 18 now twenty-twos 2 is equal to 20 factorial divided by 2 factorial times 18 factorial which can be simplified to 20 times 19 over 2 times 1 so the probability that X is equal to 2 is 20 times 19 over 2 times 1 times not point 3 squared times not point 7 to the power of 18 which is naught point naught 2 7 8 to 4 decimal places the next part of the question asks for the probability of getting more than 340 bolts what we're still dealing with the same random variable X which has the binomial distribution with 20 trials and naught point 3 is the probability of success but this time it's asking us for the probability that X is greater than 3 well this is a probability that we'll want to work out using the tables but we can't look up the probability that X is greater than 3 directly instead we need to realize that the chance of X being greater than 3 is 1 minus the probability that X is less than or equal to 3 because the opposite of being greater than 3 is being less than or equal to 3 we can find the property that X is less than or equal to 3 by looking in the tables where n is equal to 20 P is equal to not point 3 and then following the row long from 3 to find the probability which is not 0.107 1 so the chance of having more than 340 bolts is one takeaway not point 107 1 which is not 0.8929 the third part of the question introduces a new situation now we're told that bolts are sold in bags of 20 and that John buys 10 bags the question is asking us for the probability that exactly six of these bags contain more than three faulty bolts now at this point we need to introduce a new random variable we'll call it Y and make it equal to the number of bags with more than three faulty bolts this has a completely different distribution this time why has the binomial distribution with ten trials and not 0.8929 as the of relative success 10 because there are 10 bags that John buys and not 0.8929 because that was the answer to the previous question the probability that an individual bag has more than 3 faulty bolts now the question is asking us for the probability that y is equal to 6 and this time the formula tells us that that's 10 choose 6 times not 0.8929 ^ 6 times not 0.107 1 ^ 4 10 - 6 is 10 factorial over 6 factorial times 4 factorial which is 10 times 9 times 8 times 7 over 4 times 3 times 2 times 1 so now I've got a sum that we can do to calculate the probability that y is equal to 6 and if you do it you get the answer naught point naught 1 4 0 to 4 decimal places and that's the answer to the question the next question I want to look at is from the January 2002 paper and it's about organic food it tells us that the probability of a diner in a restaurant choosing organic food is 40 percent and that we're looking at a sample of 20 diners the first question asks for a suitable model to describe the number of diners who choose organic food and then it wants to know the probability that that number is greater than 5 and less than 15 well the distribution of the number of people who choose organic food is binomial with 20 trials and naught point 4 is the probability of success 20 trials because there are 20 diners and not 0.4 because that's the probability of somebody choosing to eat organic food the question wants to know the probability that X is greater than 5 but less than 15 and this is probably something that we should think carefully about for X to be greater than 5 and less than 15 means that it's six seven eight and so on up to 14 and note that we can get those numbers by taking all the numbers from 14 and below and then subtracting the ones from 5 and below so the probability that X is greater than 5 and less than 15 is the probability that X is less than or equal to 14 take away the probability that X is less than or equal to 5 we can look up both these probabilities in the tables we need to find the table where n is 20 P is not 0.4 and then following the rows from 14 which gives us not 0.99 8 4 & 5 which gives us not 0.125 6 so the probability that we're looking for is not point 9 9 8 4 take away not point 1 2 5 6 which is not point 8 7 2 8 the last part of the question asks for the mean and the standard deviation of the number of diners who choose organic food but remember that X has the binomial distribution with 20 trials and not point 4 is the probability of success so the mean will be 20 times not point 4 which is 8 the variance will be 20 times not 0.4 times not 0.6 which is 4 point 8 and therefore the standard deviation which is the square root of the variance will be the square root of 4 point 8 which is 2 point 1 9 2 3 significant figures the last question I want to look at is in the June 2003 paper which is about biased dice the first part of the question asks us to write down the conditions under which the binomial distribution will be a suitable model and you should know that there are four conditions so it's not surprising that this is for four marks the first condition is that the number of trials is fixed the second is that each trial should have the same two possible outcomes the third is that the trials must be independent and the fourth is that the probability of success must be the same in each trial moving on to the next part of the question it tells us that when the die is thrown the number 5 is twice as likely to appear as any other number although the other faces are all equally likely to appear we've got to find the probability first of all that the first 5 will occur on the sixth the throw ok well first of all we need to work out the probability of getting each number on the die we can say that the probability of getting the numbers 1 2 3 4 & 6 is P and the probability of getting a 5 is 2 P but now we know that these probabilities must add up to 1 so we can say that 7 P is equal to 1 and P as 1/7 that means that the probability of getting a 5-2 p is equal to 2/7 now the probability of getting the first 5 on the sixth throw is the probability of getting 5 non fives followed by a 5 now that's just going to be 5/7 to the power of 5 the probability of getting 5 non fives in a row times 2/7 the probability that the last throw is a 5 and that probability is equal to naught point naught 5 3 1 2 3 significant figures the last part of the question is asking us for the probability that out of the first eight throws exactly three are fives so for this let X be the number of fives it follows that X will have the binomial distribution with eight trials and 2/7 as the probability of success eight because there are eight throws and two sevens because that's the probability of getting a five we're being asked that X is equal to three and we know from the formula that that would be eight choose three times 2/7 cubed times 5/7 to the power of five eight choose three is eight factorial over 3 factorial times 5 factorial and that's 8 times 7 times 6 over 3 times 2 times 1 so that the sum that we have to do is 8 times 7 times 6 over 3 times 2 times 1 times 2/7 cubed times 5/7 to the power of 5 which is not 0.242 9 okay thank you for watching this video of exam questions it has to be said that exam questions on the binomial distribution often lead on to other topics and so these questions have not been particularly representative nevertheless I hope you found this useful revision

Channel: MrNichollTV

Views: 40,172

Keywords: Mathematics, Statistics, A-Level, S2, Edexcel, Binomial

Id: cw7ZTEDSdcI

Length: 11min 59sec (719 seconds)

Published: Wed Sep 26 2012