Understanding the Binomial Distribution: Key Concepts and
School
Union High School**We aren't endorsed by this school
Course
MATH STATISTICS
Subject
Statistics
Date
Dec 10, 2024
Pages
4
Uploaded by fitapis902
The binomial distribution –Worksheet A Section 12 –The binomial distribution VCE Mathematical Methods –Units 3 and 4 VCE Mathematical Methods –Units 3 and 4 Page 1of 4Section 12 –The binomial distribution OLVER EDUCATION12WORKSHEET A Name: SOLUTIONS QUESTION 1[WA,2012] .TA.A new treatment for back pain is being tested. A trial group consists of 100 randomly chosen patients with back pain. There is a 25% chance that any one of these patients will report an improvement after one month if no treatment is given. Let 𝑋denote the number of patients who will report an improvement after one month, assuming that no treatment is given. (a)Is the random variable 𝑋discrete or continuous? 1 mark (b)State, and fully define, the probability distribution of 𝑋. 1 mark (c)Calculate the mean and standard deviation of 𝑋. 2 marks (d)What is the probability that 35 or more of the patients in the trial group will report an improvement after one month, assuming that no treatment is given? State your answer correct to four decimal places. 2 marks (e)Now suppose that each patient in the trial group is given the new treatment and that 35 of them report an improvement after one month. Is this strong evidence that the treatment is effective? Justify your answer. 2 marks QUESTION 2[TAS,2014] .TF.Seven standard dice are rolled. Write down an expression for the probability that at least one die shows a six. 2 marks
OLVER EDUCATIONVCE Mathematical Methods –Units 3 and 4 Page 2of 4Section 12 –The binomial distribution QUESTION 3[WA,2013] .TA.A computer store room contains 25 computers. Of these, 20 are working and five need to be repaired. An order for five computers is received. The computer technician selects five computers at random from the store room. (a)What is the probability that all five are working? 2 marks (b)What is the probability that more than one needs to be repaired? 2 marks (c)The technician checks the first three of the five computers and finds that they are all working. What is the probability that all five are working? 2 marks QUESTION 4[WA,2016S] .TF.Harry fires an arrow at a target 𝑛times. The probability, 𝑝, of Harry hitting the target is constant and all shots are independent. Let 𝑋be the number of times Harry hits the target in the 𝑛attempts. The mean of 𝑋is 32 and the standard deviation is 4. (a)State the distribution of 𝑋. 1 mark (b)Determine 𝑛and 𝑝. 4 marks
OLVER EDUCATIONVCE Mathematical Methods –Units 3 and 4 Page 3of 4Section 12 –The binomial distribution QUESTION 5[TAS,2012] .TA.A fair coin is tossed three times and the number of heads (𝐻) and number of tails (𝑇) are recorded. (a)List the sample space. 1 mark (b)State the probability of getting exactly two heads. 1 mark QUESTION 6[TAS,2012] .TF.A binomial distribution consists of 100 trials. The variance of the number of successes is 21. Determine the two possible values for the probability of success on each trial. 3 marks QUESTION 7[TAS,2012] .TA.Assume that the probability of a randomly chosen person having their birthday in January is 112. If 𝑁people are chosen at random, find the smallest integer value of 𝑁so the probability that at least one person has a birthday in January exceeds 0.75. 3 marks
OLVER EDUCATIONVCE Mathematical Methods –Units 3 and 4 Page 4of 4Section 12 –The binomial distribution QUESTION 8[WA,2015] .TA.The probability that a potential buyer viewing a property buys it is 0.1. Assume people viewing the property decide independently of each other whether to buy the property or not, and that only one person views the property at a time. If the person does not buy the property, only then will a new person be allowed to view the property. (a)What is the probability that the property is sold to the second person viewing it? 2 marks (b)What is the probability that more than two people view the property before it is sold? 2 marks (c)Four people are scheduled to view the property. What is the probability that one of them buys it? 2 marks QUESTION 9[TAS,2014] .TA.A box contains 100 marbles of which 50 are yellow and 50 are green. A marble is drawn, its colour is recorded and then it is put back into the box. This is repeated until 20 marbles have been drawn and their colours recorded. (a)What is the probability of getting an equal number of yellow and green marbles? 1 mark (b)Explain why the probability of getting more yellow marbles than green marbles is exactly the same as the probability of getting fewer yellow marbles than green marbles. 2 marks (c)Using your answers to both parts (a)and (b), determine the probability of getting fewer yellow marbles than green marbles. State your answer correct to three decimal places. 2 marks