MATH500 mathematical concepts

Assignment 1

Instructions:

Please attach this sheet to the front of your assignment.

The assignment must be dropped in the Assignment Boxes on WT Level 1 (City students) or the MB foyer (AUT South).

Answer all questions and show your working. No working = no marks.

This is an individual assignment. The point of this assignment is for you to go through the process of discovery for yourself. Copying someone else’s work will not achieve this. Plagiarism has occurred where a person effectively and without acknowledgement presents as their own work the work of others. That may include published material, such as books, newspapers, lecture notes or handouts, material from the internet or other students’ written work. It also includes computer output.

The Department of Mathematical Sciences regards any act of cheating including plagiarism, unauthorised collaboration and theft of another student’s work most seriously. Any such act will result in a mark of zero being given for this part of the assessment and may lead to disciplinary action.

Please sign to signify that you understand what this means, and that the assignment is your own work.

Signature: ........................................

Question 1. [8 marks]

Let A, B, and C be subsets of a universal set U and suppose n(A) = 22, n(B) = 25, n(C) = 30, n(A ∩ B) = 9, n(A ∩ C) = 12, n(B ∩ C) = 10 and n(A ∩ B ∩ C) = 6.

• Find n(A ∪ C).
• Find n(A ∩ C⁰).
• Find n(A ∩ B ∩ C⁰).
• Find n(A ∪ B ∪ C).

Hint: Use a Venn diagram.

Question 2. [15 marks]

A spinner has four equal sectors and a number is written on each sector; 1,2, 3 and 4. A two-digit number is formed by spinning two times. The number on the first spinning makes the first digit and the number on the second spinning makes the second digit. For example, 2 on the first spinning and 1 on the second spinning make the number 21.

• Give the sample space S for the experiment.
• Consider the following events : E = even number; F = number smaller than 30; G = integer multiple of 3. Give the subset of outcomes in S that defines each of the events E, F, and G. (c) Describe the following events in terms of E, F, and G and find the probabilities for the events.
  • getting an even integer less than 30.
  • getting an odd number or an integer multiple of 3.
  • getting an odd number greater than or equal to 30 that is not a multiple of 3.
  • an even number greater than or equal to 30 that is not a multiple of 3.

(d) Are E and F mutually exclusive events? Give a reason for your answer.

Question 3. [10 marks]

• A pair of identical fair dice is rolled. What is the probability that the sum of the numbers is more than 6?
• A pair of identical fair dice is rolled. What is the probability that one of the numbers is more than 4?
• A pair of identical fair dice is rolled. What is the probability that the sum of the numbers is more than 6, if it is known that one of the numbers is a 4?
• A pair of fair dice, one red and one green, is rolled. What is the probability that the sum of the numbers is more than 6, if it is known that the red one shows a 4?

Question 4. [20 marks]

A survey shows that 80% of a population has been vaccinated against the flu, but 5% of the vaccinated population gets the flu anyway. In total 10% of the population gets the flu. Let V be the event that a randomly selected person in the population has been vaccinated and F the event that a randomly selected person in the population gets the flu.

• Present the information using either a Venn diagram or a table or both.
• Estimate the following probabilities and describe the probabilities in words.
  • P(F⁰)
  • P(F ∩ V )
  • P(F ∩ V ⁰)
  • P(F|V )
  • P(V |F)
• Does this survey provide evidence that the events F and V are not independent? Give a reason for your answer.
• Which conditional probabilities will you calculate to explain whether vaccination should be encouraged?

Question 5. [5 marks]

The key for a electronic lock is five digits long.

• How many different keys are there?
• How many different keys are there, if the first digit may not be 0?
• How many different keys are there, if the same digit may not be repeated five times?

Question 6. [10 marks]

There are 101 coins in a box, three of which are fake and 98 are genuine. Eleven randomly selected coins are tested.

• In how many ways can eleven coins be selected from the 101 coins in the box? Give the answer in factorial notation.
• What is the probability that at least one of the fake coins is selected?

Question 7. [12 marks]

• A computer “byte” consists of eight “bits”, each bit being either a 0 and a 1. If characters are represented with a code that uses a byte for each character, how many different characters can be represented?
• Some written languages, like Chinese and Japanese, use tens of thousands of different characters. If a language uses roughly 50 thousand characters, a computer code for this language would have to use how many bytes per character?
• A bag contains three red marbles, three blue ones and three green ones. How many sets of four marbles are there in which exactly two are red marbles? Assume that all the marbles are distinguishable for one another.
• A bag contains three red marbles, three blue ones and three green ones. How many sets of four marbles are there in which at least one is green? Assume that all the marbles are distinguishable for one another.

Question 8. [20 marks]

Data compiled by the Highway Patrol Department regarding the use of seat belts by drivers in a certain area is shown in the accompanying table:

• Present the information in the table, on a probability tree.
• What is the probability that a random driver in the area is stopped for a speed violation?
• If a driver in that area is stopped for a speed violation, what is the probability that he or she will have a seat belt on?
• If a driver in that area is stopped for a moving violation, what is the probability that he or she will not have a seat belt on?
• Are the events “stopped for a speeding violation” and “using a seat belt” independent events?