Download Review Probability - IB

1.
Consider the data set {k − 2, k, k +1, k + 4}, where k
(a)
.
Find the mean of this data set in terms of k.
(3)
Each number in the above data set is now decreased by 3.
(b)
Find the mean of this new data set in terms of k.
(2)
(Total 5 marks)
2.
A recruitment company tests the aptitude of 100 applicants applying for jobs in engineering.
Each applicant does a puzzle and the time taken, t, is recorded.
The cumulative frequency curve for these data is shown below.
IB Questionbank Mathematics Higher Level 3rd edition
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Using the cumulative frequency curve,
(a)
write down the value of the median;
(1)
(b)
determine the interquartile range;
(2)
(c)
complete the frequency table below.
Time to complete puzzle in seconds
Number of applicants
20 < t ≤ 30
30 < t ≤ 35
35 < t ≤ 40
40 < t ≤ 45
45 < t ≤ 50
50 < t ≤ 60
60 < t ≤ 80
(2)
(Total 5 marks)
3.
Let A and B be events such that P(A) = 0.6, P(A  B) = 0.8 and P(A  B) = 0.6.
Find P(B).
(Total 6 marks)
4.
The probability distribution of a discrete random variable X is defined by
P(X = x) = cx(5 − x), x = 1, 2, 3, 4.
(a)
Find the value of c.
(3)
IB Questionbank Mathematics Higher Level 3rd edition
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(b)
Find E(X).
(3)
(Total 6 marks)
5.
(a)
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)
(b)
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.
(i)
If E(Y2) = 5.5, find the value of m.
(ii)
Find the probability that Levi makes exactly three mistakes.
(5)
(c)
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)
6.
John removes the labels from three cans of tomato soup and two cans of chicken soup in order
to enter a competition, and puts the cans away. He then discovers that the cans are identical, so
that he cannot distinguish between cans of tomato soup and chicken soup. Some weeks later he
decides to have a can of chicken soup for lunch. He opens the cans at random until he opens a
can of chicken soup. Let Y denote the number of cans he opens.
Find
(a)
the possible values of Y,
(1)
(b)
the probability of each of these values of Y,
(4)
(c)
the expected value of Y.
(2)
(Total 7 marks)
7.
(a)
The independent random variables X and Y have Poisson distributions and Z = X + Y. The
means of X and Y are  and  respectively. By using the identity
IB Questionbank Mathematics Higher Level 3rd edition
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P Z  n  
n
 P  X  k  P Y  n  k 
k 0
show that Z has a Poisson distribution with mean ( + ).
(6)
(b)
Given that U1, U2, U3, … are independent Poisson random variables each having mean m,
n
use mathematical induction together with the result in (a) to show that
U
r
has a
r 1
Poisson distribution with mean nm.
(6)
(Total 12 marks)
8.
The heights, x metres, of the 241 new entrants to a men’s college were measured and the
following statistics calculated.
 x  412.11,  x
(a)
2
 705.5721
Calculate unbiased estimates of the population mean and the population variance.
(3)
(b)
The Head of Mathematics decided to use a χ2 test to determine whether or not these
heights could be modelled by a normal distribution. He therefore divided the data into
classes as follows.
Interval
Frequency
x  1.60
1.60  x  1.65
1.65  x  1.70
1.70  x  1.75
1.75  x  1.80
x  1.80
5
34
70
72
48
12
(i)
State suitable hypotheses.
(ii)
Calculate the value of the χ2 statistic and state your conclusion using a 10 level of
significance.
(12)
(Total 15 marks)
IB Questionbank Mathematics Higher Level 3rd edition
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9.
The number of telephone calls received by a helpline over 80 one-minute periods are
summarized in the table below.
(a)
Number of calls
0
1
2
3
4
5
6
Frequency
9
12
22
10
11
8
8
Find the exact value of the mean of this distribution.
(2)
(b)
Test, at the 5 level of significance, whether or not the data can be modelled by a
Poisson distribution.
(12)
(Total 14 marks)
10.
A factory makes wine glasses. The manager claims that on average 2 of the glasses are
imperfect. A random sample of 200 glasses is taken and 8 of these are found to be imperfect.
Test the manager’s claim at a 1 level of significance using a one-tailed test.
(Total 7 marks)
11.
Anna has a fair cubical die with the numbers 1, 2, 3, 4, 5, 6 respectively on the six faces. When
she tosses it, the score is defined as the number on the uppermost face. One day, she decides to
toss the die repeatedly until all the possible scores have occurred at least once.
(a)
Having thrown the die once, she lets X2 denote the number of additional throws required
to obtain a different number from the one obtained on the first throw. State the
distribution of X2 and hence find E(X2).
(3)
(b)
She then lets X3 denote the number of additional throws required to obtain a different
number from the two numbers already obtained. State the distribution of X3 and hence
find E(X3).
(2)
IB Questionbank Mathematics Higher Level 3rd edition
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(c)
By continuing the process, show that the expected number of tosses needed to obtain all
six possible scores is 14.7.
(5)
(Total 10 marks)
12.
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.
(c)
(i)
Find the probability that at least three telephone calls are received by Euler College
in each of two successive one-minute intervals.
(ii)
Find the probability that Euler College receives 15 telephone calls during a
randomly selected five-minute interval.
(8)
(Total 21 marks)
13.
Over a one month period, Ava and Sven play a total of n games of tennis.
The probability that Ava wins any game is 0.4. The result of each game played is independent
of any other game played.
Let X denote the number of games won by Ava over a one month period.
IB Questionbank Mathematics Higher Level 3rd edition
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(a)
Find an expression for P(X = 2) in terms of n.
(3)
(b)
If the probability that Ava wins two games is 0.121 correct to three decimal places, find
the value of n.
(3)
(Total 6 marks)
14.
The lifts in the office buildings of a small city have occasional breakdowns. The breakdowns at
any given time are independent of one another and can be modelled using a Poisson Distribution
with mean 0.2 per day.
(a)
Determine the probability that there will be exactly four breakdowns during the month of
June (June has 30 days).
(3)
(b)
Determine the probability that there are more than three breakdowns during the month of
June.
(2)
(c)
Determine the probability that there are no breakdowns during the first five days of June.
(2)
(d)
Find the probability that the first breakdown in June occurs on June 3rd.
(3)
(e)
It costs 1850 euros to service the lifts when they have breakdowns. Find the expected cost
of servicing lifts for the month of June.
(1)
(f)
Determine the probability that there will be no breakdowns in exactly 4 out of the first 5
days in June.
(2)
(Total 13 marks)
IB Questionbank Mathematics Higher Level 3rd edition
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15.
Only two international airlines fly daily into an airport. UN Air has 70 flights a day and IS Air
has 65 flights a day. Passengers flying with UN Air have an 18 probability of losing their
luggage and passengers flying with IS Air have a 23 probability of losing their luggage. You
overhear someone in the airport complain about her luggage being lost.
Find the probability that she travelled with IS Air.
(Total 6 marks)
16.
A company produces computer microchips, which have a life expectancy that follows a normal
distribution with a mean of 90 months and a standard deviation of 3.7 months.
(a)
If a microchip is guaranteed for 84 months find the probability that it will fail before the
guarantee ends.
(2)
(b)
The probability that a microchip does not fail before the end of the guarantee is required
to be 99. For how many months should it be guaranteed?
(2)
(c)
A rival company produces microchips where the probablity that they will fail after 84
months is 0.88. Given that the life expectancy also follows a normal distribution with
standard deviation 3.7 months, find the mean.
(2)
(Total 6 marks)
IB Questionbank Mathematics Higher Level 3rd edition
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