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PRIFYSGOL CYMRU/UNIVERSITY OF WALES
ABERYSTWYTH
INSTITUTE OF MATHEMATICAL AND PHYSICAL SCIENCES
SEMESTER 1 EXAMINATIONS, JANUARY/FEBRUARY 2008
MA103 10 – Probability
Time allowed – 2 hours
- Full marks will be given for complete answers to ALL questions in Section A and to
THREE questions in Section B. In Section B credit will be given for the best three questions.
- Calculators are permitted, provided they are silent, self-powered, without communications facilities, and incapable of holding text or other material that could be used to give a candidate an unfair advantage. They must be made available on request for inspection by invigilators, who are authorised to remove any suspect calculators.
Information
You may quote the results that for |x|<1,
( )
( )
2 3
(^2 )
x x x (^) x
x x x (^) x
etc.
MA10310:Probability January/February 2008 Page 2 of 6
Section A
1 On three days we measure the temperature Ti °C for i = 1,2,3. For i = 1,2, define the event Ai to be that the temperature on the i-th day is below freezing. Match up the following four events with the correct verbal description (i) A 1 ∩ A 2 ∩ A 3 (a) The three days are not all below freezing (ii) A 1 ∪ A 2 ∪ A 3 (b) All three days are above freezing (iii) A 1 ∩ A 2 ∩ A 3 (c) At least one day is above freezing (iv) A 1 ∪ A 2 ∪ A 3 (d) It is not true that any of the three days is below freezing Which of (a)-(d) describe the same event? [5]
2 A point P with coordinates (X,Y) is chosen at random inside a square whose vertices are the origin, (0,2), (2,0) and (2,2). Find the chances that (i) Y < 3X; (ii) the distance from P to the origin is less than 1; (iii) P is nearer to (1,1) than to any vertex. [5]
3 From the probabilities displayed in the following Venn diagram:
write down the values of (i) P ( A ∩B ) , (ii) P ( A | B ) and (iii) P( B | A).
(iv) What are the odds against A and B both occurring? (v) Are A and B independent? [5]
4 A poker hand contains five cards drawn, without replacement, from a standard deck of 52 playing cards. Calculate the probabilities that the hand contains (i) all Clubs; (ii) no black cards; (iii) two Hearts and one Diamond. [5]
0.3 0.2 0.^2
A B
MA10310:Probability January/February 2008 Page 4 of 6
Section B
Credit will be given for the best THREE answers from this section.
9B State the Axioms of Probability. Assuming only the Axioms of Probability, prove that P(A∪B) = P(A) + P(B) – P(A∩B) and write down an expression for P(A∪B∪C) in terms of the probabilities of A,B,C and their intersections. A and B and C are events in a sample space and it is known that
- A and C are mutually exclusive;
- A is independent of B;
- the odds against A occurring are 4 to 1
- the chance that B is the only event to occur is 0.33;
- C is twice as likely to occur as A ; (i) Write each of these statements as a probability equation. (ii) Given further that, conditional on B occurring, A and C are equally likely, find P( A | B ∪C).
[3]
[5]
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10B (a)^ What are the odds against the scores on 5 fair dice all being different?
(b) Four cards have the letter I printed on them, four have the letter S, two have the letter P and one further card has the letter M. What is the probability that if the eleven cards are arranged randomly in a line they will spell the word MISSISSIPPI?
(c) If I count the number of times a fair coin is thrown up to and including the throw on which the third head is obtained, how likely is it that this number is divisible by 3?
(d) Sam arrives at the bus stop at some time between 8.30am and 8.45am in the morning, and his bus arrives between 8.33am and 8.43am. Making suitable assumptions (which should be stated), how often does Sam catch the bus?
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MA10310:Probability January/February 2008 Page 5 of 6
11B (a) Of the motor traffic passing along Llanbadarn Road at certain times during the day, 60% are private vehicles, 20% are goods vehicles, 15% are buses and 5% are taxis. It is known that a half of private vehicles and all buses are diesel powered. 60% of taxis and 80% of goods vehicles use diesel fuel. What proportion of all vehicles use diesel? What are the odds against a vehicle being a goods vehicle (i) if nothing is known about its engine?; (ii) if it is known to be diesel powered? (b) Define the conditional probability P(A|B). An event B is said to favour an event A if P(A|B) > P(A). Prove that if B favours A then A favours B. Does B favourA? Justify your answer.
[10]
[10]
12B Define the expected value, E[N], of a discrete random variable N whose probability mass function (pmf) is pn. (a) Computer games are delivered in boxes of 20 to a shop. The manager chooses four at random from each box to be tested. In how many ways can the manager choose the four? Suppose there are 3 faulty games in the box. What is the probability that the manager finds (i) all three? (ii) none? Calculate (i) the pmf of F, the number of faulty games which the manager finds; (ii) the expected number of faulty games. (b) The manager decides to return a box if either there are 2 or more faulty games in the sample of four; or if there is 1 faulty game in the sample of four and a second random sample of four reveals any more faulty games. What is the probability that this box, containing 3 faulty games, is returned? (c) The manager applies the same procedure to 12 of these boxes, each with the same probability of return. Find (i) the probability that exactly one box is returned; (ii) the expected number of boxes that will be returned.
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