math assignment on ( probabilities )

  

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MAT

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CPE ASSIGNMENT 5,

2

013

Place your assignment solutions in the appropriate pigeonhole in the boxes on the third level of the
Mathematics Building before 12.00noon on Thursday 16ththth May.

Each page of your solutions must carry (1) your name, (2) your demonstrator’s name,
and (3) the day and time of your first Practice Class of the week.

In submitting your work, you are consenting that it may be copied and transmitted by the
University for the detection of plagiarism. You must start your assignment solutions with
the following Statement of Originality, signed and dated by you:

“This is my own work. I have not copied any of it from anyone else.”

The assignments are designed to help you master the concepts in this subject and also for you to
develop your mathematical communication skills. Please note that often it is not the final answer
that is important but your mastery of the required techniques and the way you communicate your
ideas and your approach to the problems.

1. (a) Simplify the following expression:

x−12(x3 + y4(xz)−7)

x−20y4z−6
+

x

z
.

(b) Consider the function f : R → R given by

f(x) =

{
x if x ∈ [0, 3],
0 otherwise.

(i) Sketch the graph of f.

(ii) Use

∫ 2
−∞

f(x) dx =

∫

0

−∞

f(x) dx +

∫ 2
0

f(x) dx to calculate

∫ 2
−∞

f(x) dx.

(iii) Calculate

∫ ∞
−∞

f(x) dx.

2. (a) In this question we consider the function g(x) = x3 − 3×2 − 9x.

(i) By factorising and using the quadratic formula, find the three solutions to the equation
g(x) = 0, thus finding the points where g(x) intersects the horizontal axis. Give exact
answers.

(ii) Calculate g′(x) and find the x value of all stationary points by solving g′(x) = 0.

(iii) Calculate g(x) for the points found in (b) to get the y co-ordinate for each stationary
point.

(iv) Calculate g′′(x) and use the second derivative test (see assignment 4) to classify the
stationary points.

(v) Solve g′′(x) = 0 to find the x value of the point of inflection and then calculate its
corresponding y value.

(vi) Use (a) to (e) to make a clear and neat sketch of the graph of g. Make sure your
graph is large enough for you to clearly label the x and y co-ordinates of stationary
points and points of inflection on the axes as well as labelling intercepts.

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Questions 3 and 4 concern the probability part of the course.

3. Say that two events A and B are known to have probability P (A) = 3
5

and P (B) = 1
3

respec-
tively. Suppose further that P (A′ ∪B) = 3

5
also.

(a) Calculate P (A ∩ B) using the axioms and basic properties of probability. (See Practice
Class 8A.)

(b) Are A and B independent events? (See Practice Class 9A.)

4. (a) (See Probability Practice Class 8.) A bag contains 4 balls, b1, b2, b3, b4. Ball b1 is blue,
balls b2 and b3 are red and ball b4 is white. First one, then (without replacement) a second
ball is taken from the bag. The balls are chosen randomly: at each selection, the available
selections have equal probability of occurring.

(i) If the experiment asks us to observe the colour of the balls, what is the outcome
space, Ω1? (Give brief explanation to any notation you introduce.)

(ii) If the experiment asks us to observe the names of the balls (b1, b2, b3 and b4), there
are 12 possible outcomes, and the elementary events have equal probabilities. Write
down the outcome space, Ω2, in this case. (Give brief explanation to any notation you
introduce.)

(iii) Let A be the event that at least one of the balls is red and B be the event that the
balls have different colours. Identify each of the events A, B and A ∩ B as subsets of
Ω1 and as subsets of Ω2 and then state the probabilities P (A), P (B) and P (A∩B).

(iv) Use the definition of conditional probability to find the the probability that one of the
balls is red, given that the two balls selected have different colours. (Your solution
should note where the definition of conditional probability is being used.)

(b) (See Probability Practice Class 9.) Suppose that around 99% of all accessible websites
contain no illegal content. Suppose also that a program designed to censor illegal websites,
correctly censors 95% of all websites containing illegal content but incorrectly censors
around 5% of legitimate websites (containing no illegal content). What is the probability
that a website contains illegal content given that the program censors it? Be careful to
explain any notation you introduce, and state any named theorem that you use.

Questions 5 and 6 concern the calculus part of the course.

5. Evaluate the following integrals using substitution.

(a)

∫ π
4

0

sin
(

ln(cos(x))
)

tan(x) dx [This one’s not as bad as it looks! Check the table of

common indefinite integrals in appendix B of the calculus notes.]

(b)

∫ 1
0

2×2
√

1 −x6
dx. [Hint: Use the substitution y = x3. For a similar example, see Question 2

of Calculus Practice Class 8.]

6. (a) Use integration by parts to evaluate

∫ 5
1

2x ln(5x) dx.

(b) Let v = cos2(x). Calculate
dv

dx
.

(c) Use your answer to (b) and the By Parts rule to evaluate

∫ π
4
0

−2 cos(x) sin(x) tan(x) dx.

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