Download Department of Mathematics Centre for Foundation Studies, IIUM

Department of Mathematics
Centre for Foundation Studies, IIUM
Semester III, 2013/2014
SHF1124 (MATHEMATICS II)
TUTORIAL 5
CHAPTER 5: DISCRETE PROBABILITY DISTRIBUTIONS
Section
Page Number
Questions
5-1: Probability Distribution
p.316
2, 4, 7 (a)(b)(c), 9(a)
5-2: Mean, Variance and Expectation p.325
8, 9, 11
5-3: The Binomial Distribution
p.334
7, 16, 18
5-4 :The Poisson Distribution
p. 343
8, 12, 13
Review Exercises
p. 346
7, 13
*Required Textbook: - Salina Mohin et al , Mathematics & Statistics for Pre-University,
McGraw-Hill Education (Malaysia) Sdn. Bhd.(2013)
Errata for questions from the textbook:
Section
Question
5.1
2(c) Answer : 0.08
9(a) Answer : 29/33
5.3
18(b) 0.4503
Review 13. Answer : 0.9834, 0.5591,
149
Section
Question
5.2
8. In a game, a player tosses
THREE..
11. In a game, a player is
given 3 trials…
5.4
13(a) Answer : 0.5830
EXTRA QUESTIONS
1. A newsagent sells telephone cards at RM1, RM2, RM4, RM10 or RM20. The value,
in Ringgits, of a telephone card sold may be regarded as a random variable , X,
with the following probability distribution.
x
1
2
4
10
20
P(X = x)
p
2p
2q
q
0.07
(a) Find the value of p and q, given that 3P(X > 2) = 2P(X ≤ 2)
(b) Find the mean and standard deviation of X.
(c) Calculate the probability that the value of the next telephone card sold is less
than RM4.
2. A box contains five balls labeled 1 , 2 , 3 , 4 and 5. A player draws a ball from the
box randomly. If the number on the ball drawn is 3 ,4 or 5 , the player’s score is
that number. If the number on the ball is 1 or 2 , the player can draw a second
ball without replacing the first ball, and his score is the sum of the numbers on
the two balls. Let X be the random variable that represents the score.
(a) Tabulate the probability distribution for X.
(b) Draw a graph for the distribution.
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3. A bag contains 3 red balls and 2 black balls. Balls are drawn at random from the
bag one by one, without replacement, until a black ball is obtained. The number
of draws required to obtain a black ball is represented by a random variable X.
(a) Construct the probability distribution table for X.
(b) Find the variance of X.
4. There are 15 parking lots along a road. The probability of any one of the lots still
empty at 9.00 a.m. on a particular Monday is 0.3. Find, correct to 4 decimal
places, the probability that, on that Monday morning,
(a) between 5 and 13 parking lots are filled.
(b) at most 10 parking lots are still not filled.
5. Analysis of records over a long period reveals that 4% of the scholarship holders
will break their bonds before their term of service expires.
(a) Calculate the probability that in a random sample of 8 scholarship holders
exactly 2 will break their bonds.
(b) Find the minimum number of scholarship holders that should be sampled if
there is to be a probability of more than 0.95 that at least 1 scholarship
holder will break his bond.
(c) The government awarded 350 scholarships in a particular year. Find, using a
suitable approximation, the probability that between 15 and 20 (inclusive)
scholarship holders will break their bonds.
6. Assume that the number of customers who enters an antique shop per hour has
a Poisson distribution with mean 1.7.
(a) Find the probability that at least 3 customers will enter the shop in a
particular hour.
(b) Each morning, a worker of the shop needs 30 minutes to get ready after
opening the shop. Find the probability that he will not be interrupted by a
customer in that period of time.
(c) Find the probability that from 10.00 a.m to 1.00 p.m on a particular day, at
least 3 customers entered the shop each hour.
90% of the customers who enter the shop do not buy anything. If 10 customers
enter the shop on a particular day, find the probability that less than 3 customers
buy something.
7. In a certain shopping complex, the bookshop SPH opens for business everyday
and on average, 3 copies of the book “Resepi Harian” are sold in every 4 days
and the number of books sold are found to follow a Poisson distribution.
(a) Find the probability that no copy of the book is sold on a particular day,
giving your answer correct to 4 decimal places.
(b) Find the least value of M so that the probability of less than M copies of the
book are sold on 4 consecutive days is more than 0.4.
(c) Find the probability that, in a particular week, there are less than 3 days
with no sale of the book at all.
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8. In a delivery of microchips, it is known that the number of defective is 2 out of 10.
(a) If 15 microchips are delivered, calculate the probability that exactly 11
microchips are good.
(b) Suppose in another shipment of microchips, the probability of defective is
0.01. If a sample of 300 microchips is taken from the shipment, estimate the
probability of getting 1 to 3 defective microchips.
9. In a multiple choice test, each question is given a choice of four answers, A, B, C
and D. The total number of questions is 20.
(a) If a student guesses each of the answers, calculate the probability that the
student gets at least 2 correct answers.
(b) If 10 students sit for the test and all of them guess at each of the answers,
calculate the probability that only one student do not get any correct
answer at all.
10. On the average, one out of 2000 cars breaks down on a particular express way.
In a week there are 800 cars using this express way. By using an appropriate
approximation, find the probability that in a given week
(a) no car will break down on the express way.
(b) more than 4 cars will break down on the express way.
Answer : EXTRA QUESTIONS
Question
Answer
1
(a)p=0.20 q=0.11 (b) 4.38, 5.01 (c) 0.60
2
x
3
4
5
6
7
P(X=x) 6/20 5/20 6/20 2/20 1/20
3
Var(X)=1;
x
1
2
3
4
P(X=x) 2/5 3/10 1/5 1/10
4
(a)0.8695
(b) 0.9993
5
(a) 0.035
(b)74
(c) 0.3817
6
(a) 0.2428
(b) 0.4274
(c) 0.0143
0.9298
7
(a) 0.4724
(b)3
(c) 0.2743
8
(a) 0.1876
(b) 0.5974
9
(a) 0.9757
(b) 0.0311
10
(a) 0.6703
(b) 0.0001
A person’s true wealth is the good deeds that he does in this world
~ The Prophet Muhammad (may Allah bless him and grant him peace)
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