Question bank Problems
1.
Calc (2-7,10,11,12,14)
NO Calc(1,8,9,13)
A batch of 15 DVD players contains 4 that are defective. The DVD players are selected at random, one by one, and
examined. The ones that are checked are not replaced.
(a)
What is the probability that there are exactly 3 defective DVD players in the first 8 DVD players examined?
(4)
(b)
What is the probability that the 9
th
DVD player examined is the 4
th
defective one found?
(3)
(Total 7 marks)
2.
A discrete random variable X has a probability distribution given in the following table.
x
0.5
1.5
2.5
3.5
4.5
5.5
P(X = x)
0.15
0.21
p
q
0.13
0.07
(a)
If E(X) = 2.61, determine the value of p and of q.
(b)
Calculate Var (X) to three significant figures.
(4)
3.
(2)
(Total 6 marks)
Tim throws two identical fair dice simultaneously. Each die has six faces: two faces numbered 1, two faces numbered
2 and two faces numbered 3. His score is the sum of the two numbers shown on the dice.
(a)
(i)
Calculate the probability that Tim obtains a score of 6.
(ii)
Calculate the probability that Tim obtains a score of at least 3.
(3)
Tim plays a game with his friend Bill, who also has two dice numbered in the same way. Bill’s score is the sum of the
two numbers shown on his dice.
(b)
(i)
Calculate the probability that Tim and Bill both obtain a score of 6.
(ii)
Calculate the probability that Tim and Bill obtain the same score.
(4)
(c)
Let X denote the largest number shown on the four dice.
16
.
81
(i)
Show that P(X ≤ 2) =
(ii)
Copy and complete the following probability distribution table.
(iii)
x
1
P(X = x)
1
81
2
3
2
Calculate E(X) and E(X ) and hence find Var(X).
(10)
(d)
Given that X = 3, find the probability that the sum of the numbers shown on the four dice is 8.
(4)
(Total 21 marks)
4.
In each round of two different games Ying tosses three fair coins and Mario tosses two fair coins.
(a)
The first game consists of one round. If Ying obtains more heads than Mario, she receives $5 from Mario. If
Mario obtains more heads than Ying, he receives $10 from Ying. If they obtain the same number of heads,
then Mario receives $2 from Ying. Determine Ying’s expected winnings.
(12)
(b)
5.
6.
They now play the second game, where the winner will be the player who obtains the larger number of heads
in a round. If they obtain the same number of heads, they play another round until there is a winner. Calculate
the probability that Ying wins the game.
(8)
(Total 20 marks)
There are 30 students in a class, of which 18 are girls and 12 are boys. Four students are selected at random to form a
committee. Calculate the probability that the committee contains
(a)
two girls and two boys;
(3)
(b)
students all of the same gender.
(3)
(Total 6 marks)
The continuous random variable X has probability density function
f(x) =
1
6
2
x(1 + x )
f(x) = 0
(a)
Sketch the graph of f for 0 ≤ x ≤ 2.
(b)
Write down the mode of X.
for 0 ≤ x ≤ 2,
otherwise.
(2)
(1)
IB Questionbank Mathematics Higher Level 3rd edition
1
Question bank Problems
(c)
(d)
7.
8.
Calc (2-7,10,11,12,14)
NO Calc(1,8,9,13)
Find the mean of X.
Find the median of X.
(5)
(Total 12 marks)
The speeds of cars at a certain point on a straight road are normally distributed with mean µ and standard deviation σ.
–1
–1
15 % of the cars travelled at speeds greater than 90 km h and 12 % of them at speeds less than 40 km h . Find µ
and σ.
(Total 6 marks)
The probability density function of the random variable X is given by
f(x) =
(a)
 k
, for 0  x  1

 4  x2

0,
otherwise.

Find the value of the constant k.
(5)
(b)
Show that E(X) =
6(2  3 )
.
π
(7)
(c)
Determine whether the median of X is less than
1
1
or greater than
.
2
2
(8)
(Total 20 marks)
9.
A discrete random variable X has its probability distribution given by
P(X = x) = k(x + 1), where x is 0, 1, 2, 3, 4.
(a)
Show that k =
(b)
Find E(X).
1
.
15
(3)
10.
11.
(3)
(Total 6 marks)
(a)
A box of biscuits is considered to be underweight if it weighs less than 228 grams. It is known that the
weights of these boxes of biscuits are normally distributed with a mean of 231 grams and a standard deviation
of 1.5 grams. What is the probability that a box is underweight?
(2)
(b)
The manufacturer decides that the probability of a box being underweight should be reduced to 0.002.
(i)
Bill’s suggestion is to increase the mean and leave the standard deviation unchanged. Find the value
of the new mean.
(ii)
Sarah’s suggestion is to reduce the standard deviation and leave the mean unchanged. Find the value
of the new standard deviation.
(6)
(c)
After the probability of a box being underweight has been reduced to 0.002, a group of customers buys 100
boxes of biscuits. Find the probability that at least two of the boxes are underweight.
(3)
(Total 11 marks)
The annual weather-related loss of an insurance company is modelled by a random variable X with probability
density function
 2.5(200 ) 2.5

f(x) = 
x 3.5

0,

x  200
otherwise.
Find the median.
12.
(Total 8 marks)
The distance travelled by students to attend Gauss College is modelled by a normal distribution with mean 6 km and
standard deviation 1.5 km.
(a)
(i)
Find the probability that the distance travelled to Gauss College by a randomly selected student is
between 4.8 km and 7.5 km.
(ii)
15 of students travel less than d km to attend Gauss College. Find the value of d.
(7)
At Euler College, the distance travelled by students to attend their school is modelled by a normal distribution with
mean  km and standard deviation  km.
(b)
If 10 of students travel more than 8 km and 5 of students travel less than 2 km, find the value of  and of
.
(6)
The number of telephone calls, T, received by Euler College each minute can be modelled by a Poisson distribution
with a mean of 3.5.
IB Questionbank Mathematics Higher Level 3rd edition
2
Question bank Problems
(c)
(i)
(ii)
Calc (2-7,10,11,12,14)
NO Calc(1,8,9,13)
Find the probability that at least three telephone calls are received by Euler College in each of two
successive one-minute intervals.
Find the probability that Euler College receives 15 telephone calls during a randomly selected fiveminute interval.
(8)
(Total 21 marks)
13.
A continuous random variable X has probability density function
f(x) =
It is known that P(X < 1) = 1 –
1
 0,
  ax
 ae ,
x0
x  0.
.
2
1
ln 2 .
2
(a)
Show that a =
(b)
Find the median of X.
(c)
Calculate the probability that X < 3 given that X > 1.
(6)
(5)
14.
(a)
(b)
(c)
(9)
(Total 20 marks)
Ahmed is typing Section A of a mathematics examination paper. The number of mistakes that he makes, X,
can be modelled by a Poisson distribution with mean 3.2. Find the probability that Ahmed makes exactly four
mistakes.
(1)
His colleague, Levi, is typing Section B of this paper. The number of mistakes that he makes, Y, can be
modelled by a Poisson distribution with mean m.
2
(i)
If E(Y ) = 5.5, find the value of m.
(ii)
Find the probability that Levi makes exactly three mistakes.
(5)
Given that X and Y are independent, find the probability that Ahmed makes exactly four mistakes and Levi
makes exactly three mistakes.
(2)
(Total 8 marks)
IB Questionbank Mathematics Higher Level 3rd edition
3