MATH1700 Probability and Statistics
2024–25
Problem Sheet 1
This is Problem Sheet 1, which covers revision of A-level probability. You should work through the questions on this problem sheet in advance of your tutorial in Week 1. I’m aware that some students have qualifications other than A-level mathematics. If any of the material on this sheet is unfamiliar, don’t worry too much at this stage as we will recap everything as we go along during the course.
A: Short questions
The first seven questions are short questions, which are mostly intended to be straightforward. You can check your answers with the solutions-without- working at the bottom of this sheet. It is good practice to write up complete solutions showing all your working and solutions-with-working will be available in Week 2. If you get stuck on any of these questions, you should ask for guidance in your tutorial.
A1. Suppose you toss a fair coin twice. Find the probability that:
(a) You get a head and then a tail.
(b) You get one head and one tail.
A2. A jar contains 7 red marbles, 5 blue marbles, and 8 green marbles. A marble is drawn at random. Find the probability that:
(a) The marble is red.
(b) The marble is not green.
(c) The marble is either blue or green.
A3. If ℙ(A) = 0.35, find the probability of the complement of event A, ℙ(Ac ).
A4. Two events A and B are mutually exclusive, and ℙ(A) = 0.4, ℙ(B) = 0.3. What is the probability that either A or B occurs?
A5. A bag contains 5 red balls and 3 blue balls. Two balls are drawn at random. Find the probability that both balls are red if:
(a) The balls are chosen without replacement.
(a) The balls are chosen with replacement.
A6. Two events A and B are independent, and ℙ(A) = 0.5, ℙ(B) = 0.4. Find:
(a) ℙ(A ∩ B)
(b) ℙ(A ∪ B)
A7. In a class of 30 students, 18 study mathematics, 12 study physics, and 6 study both subjects. Find:
(a) The probability that a randomly selected student studies mathematics or physics.(b) The probability that a student studies mathematics given that they study physics.
B: Long questions
The next three questions are long questions, which are intended to be harder. Long questions often require you to think originally for yourself, not just di- rectly follow procedures from the notes. You may not be able to solve all of these questions, although you should make multiple attempts to do so. Here, your answers should be written in complete sentences, and you should carefully explain in words each step of your working. Your answers to these questions – not only their mathematical content, but also how to write good, clear solutions – are likely to be the main topic for discussion in your tutorial. Solutions will be available in Week 2.
B1. In a survey of 100 people, 40 liked tea, 35 liked coffee, and 15 liked both tea and coffee. By drawing a Venn diagram, or otherwise, find the probability that a person chosen at random:(a) Likes only tea.
(b) Likes neither tea nor coffee.
(c) Likes tea given that they like coffee.
B2. A factory produces 80% of its goods in Factory A and 20% in Factory B. The probability of a defective item from Factory A is 0.05, and from Factory B, it is 0.1.
(a) Draw a tree diagram representing this situation.
(b) Calculate the probability that a randomly selected item is defective.
B3. A magician has prepared two piles of cards taken from a standard deck of cards. In the first pile he has put the 3, 6 and 7 of hearts; in the second pile he has the 2 of spades and a second numbered spade, which we denote as ∞ . A member of the audience is invited to pick one card at random from each pack and multiply the two numbers together.
(a) Draw a sample space diagram giving all possible products (in terms of ∞).
(b) Calculate the probability that the product is even in the two cases that (i) ∞ is odd and (ii) ∞ is even.
(c) The magician knows that the probability that the product is even is equal to the probability that the product is greater than 10. Find the value of ∞ .
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