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SEFYDLIAD MATHEMATEG A FFISEG
INSTITUTE OF MATHEMATICS AND PHYSICS
SEMESTER 1 EXAMINATIONS, JANUARY 2010
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.
MA11310:Statistics January 2010 Page 2 of 5
Section A
1 Simplify as concisely as possible and describe in words the event
( A^ ∪^ (^ A^ ∪^ C^ )) ∩^ (^ B^ ∩^ A) [5]
2 Four men and three women have separately bought numbered tickets for the 7 numbered seats in one row in a theatre. (a) In how many ways could the 7 people be arranged in the 7 seats? (b) What is the chance that each woman is seated between two men? [5]
3 Events A and B each have probability p and P(A∩B) = 0.2. Express in terms of p: (i) the chance that either of the two events occurs; (ii) the chance that B occurs and A does not. If the chance that neither event occurs is 0.2, find the value of p. [5]
4 If P(C) = 0.6, P(C|D) = 0.2 and P(C ∪D) = 0.9, evaluate
(i) P(D); P ( D |C). [5]
5 The discrete random variable N has probability mass function (pmf) p 1 = 0.1k, p 2 = 0.2k, p 3 = 0.3k, p 4 = 0.4k, p 5 = 0.5k. Find (i) the value of k and (ii) the expected value of N. [5]
6 The daily takings £X, of an ice cream van, have cumulative distribution function (cdf)
2 F x ( ) 1 10000 x = − for x>100 (and zero otherwise). Evaluate (i) the median; (ii) the probability density function (pdf) of X. [5]
7 The time, Z minutes, it takes to serve a customer in a shop has probability density function (pdf) f(z) = c(z – 3)^2 , for 1< z < 5. Evaluate c and verify that E[Z] = 3. [5]
8 The random variable Y has mean -8 and standard deviation 8. Evaluate (i) E[Y 2 ]; (ii) E[(0.5Y – 10)^2 ]. (^) [ 5 ]
MA11310:Statistics January 2010 Page 4 of 5
11B Define the conditional probability P(A|B). In a multiple choice examination candidates have to choose from five possible answers, one of which is correct. Tom knows the answer to any question with probability p and decides to choose at random from the 5 choices for those he does not know. Find: (i) Show that the chance that Tom answers the first question correctly is (1+4p)/5. (ii) What is the chance that Tom knew the answer to a question that he got correct? Dick, who also knows the answer with probability p decides to completely ignore any questions that he does not know the answers to, offering no answer. One mark is awarded for a correct answer, whilst c marks are deducted for any incorrect answers. No marks are awarded if no answer is offered. (iii) Find the value of c that gives Tom and Dick the same expected mark per question, showing that this does not depend on p. In answering 25 such questions independently of one another, Tom got a total mark of 15. (iv) How many questions did he get correct? (v) Guess how many questions he knew the answer to. Justify your answer.
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12B (a) During an ice hockey practice session, Arwel takes penalty shots until he succeeds in scoring. Independently of previous results, his success probability is 0.6. (i) Calculate the probability that he will need at least four attempts to score. (ii) Find the smallest n for which there is a probability of at least 0.99 that he will succeed in n or fewer attempts. (iii) If six hockey players of the same ability as Arwel take shots independently, find the probability that four of them will score within the first three attempts. Continued
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MA11310:Statistics January 2010 Page 5 of 5
(b) A point O is chosen at random in an interval PQ of length 6cm and the length Xcm from P measured. A circle is then drawn with radius PO. Suppose its area is Acm². Express A as a function of X and show that the median of A is 9π.
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13B Define the cumulative distribution function (cdf), F(x), and the probability density function (pdf), f(x)of a random variable X.
A car headlight bulb has a random lifetime, T years, with probability density function (pdf) f t ( ) = 0.4 te−^ 0.2t^2 for t>0 (and 0 otherwise) (a) Show that the cumulative distribution function (cdf) of T is F t ( ) = 1 − e −^ 0.2t^2 , t> 0 and zero otherwise. (b) Evaluate the probability that a randomly chosen bulb lasts (i) 3 years or less; (ii) 5 years or more; (iii) for at least another 3 years, if it has already lasted for 2 years. (c) Find the probability that T exceeds twice its median.
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14B The function pi, for i = 1,2,3, … is a probability mass function (pmf). What two conditions must it satisfy? The discrete random variable K has the following probability mass function (pmf), where q is a constant such that 0 < q < 1. k 1 2 3 4 pk = P(K = k) (1 − q)^3 3 (1q − q)^23 q 2 (1 − q) q^3 (i) Verify that pk is a valid pmf. (ii) Show that E[K] = 1 + 3q.. (iii) Calculate E[K(K–1)] and hence, or otherwise, find the variance of K.
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