AIHL DISTRIBUTIONS P1 [168 marks]

1. [Maximum mark: 6] SPM.1.SL.TZ0.13

Mr Burke teaches a mathematics class with 15 students. In this class there are 6 female students
and 9 male students.

Each day Mr Burke randomly chooses one student to answer a homework question.

(a) Find the probability that on any given day Mr Burke chooses a female student
to answer a question. [1]

In the first month, Mr Burke will teach his class 20 times.

(b) Find the probability he will choose a female student 8 times. [2]

(c) Find the probability he will choose a male student at most 9 times. [3]
2. [Maximum mark: 6] SPM.1.SL.TZ0.12
Jae Hee plays a game involving a biased six-sided die.

The faces of the die are labelled −3, −1, 0, 1, 2 and 5.

The score for the game, X, is the number which lands face up after the die is rolled.

The following table shows the probability distribution for X.

(a) Find the exact value of p. [1]

Jae Hee plays the game once.

(b) Calculate the expected score. [2]

(c) Jae Hee plays the game twice and adds the two scores together.

Find the probability Jae Hee has a total score of −3. [3]

3. [Maximum mark: 6] SPM.1.AHL.TZ0.17

Mr Burke teaches a mathematics class with 15 students. In this class there are 6 female students
and 9 male students.

Each day Mr Burke randomly chooses one student to answer a homework question.

In the first month, Mr Burke will teach his class 20 times.

(a) Find the probability he will choose a female student 8 times. [2]

(b) The Head of Year, Mrs Smith, decides to select a student at random from the
year group to read the notices in assembly. There are 80 students in total in
the year group. Mrs Smith calculates the probability of picking a male student
8 times in the first 20 assemblies is 0.153357 correct to 6 decimal places.

Find the number of male students in the year group. [4]

4. [Maximum mark: 7] EXN.1.SL.TZ0.12
A disc is divided into 9 sectors, number 1 to 9. The angles at the centre of each of the sectors
u n form an arithmetic sequence, with u 1 being the largest angle.

(a) 9

Write down the value of Σ ui .

[1]
i=1

1
It is given that u 9 =
3
u1 .

(b) Find the value of u 1 . [4]

(c) A game is played in which the arrow attached to the centre of the disc is spun
and the sector in which the arrow stops is noted. If the arrow stops in sector 1
the player wins 10 points, otherwise they lose 2 points.

Let X be the number of points won

Find E(X). [2]

5. [Maximum mark: 6] 23N.1.SL.TZ2.11
Toktam works at a local bakery 5 days each week. She drives an old car to work that has a
65 % probability of starting on any given morning. The probability of the car starting on a
given morning is independent of it starting on any other morning.

(a) Find the probability that Toktam’s car starts on exactly two mornings in a
particular 5 day workweek. [2]

Toktam walks to work on mornings when her car does not start and it is not raining. Toktam
takes a taxi to work on mornings when her car does not start and it is raining.

Where Toktam lives, there is a 45 % probability of rain on any given morning, independent of
any other morning. The probability of Toktam’s car starting is independent of the weather.

(b) Find the probability that Toktam will not have to take a taxi in a particular
workweek. [4]

6. [Maximum mark: 7] 23M.1.SL.TZ1.12

On a specific day, the speed of cars as they pass a speed camera can be modelled by a normal
distribution with a mean of 67. 3 km h
−1
.

A speed of 75. 7 km h
−1
is two standard deviations from the mean.

(a) Find the standard deviation for the speed of the cars. [2]

Speeding tickets are issued to all drivers travelling at a speed greater than 72 km h
−1
.

(b) Find the probability that a randomly selected driver who passes the speed
camera receives a speeding ticket. [2]

It is found that 82% of cars on this road travel at speeds between p km h

−1
and q km h
−1

, where p < q. This interval includes cars travelling at a speed of 74 km h

−1
.

(c) Show that the region of the normal distribution between p and q is not
symmetrical about the mean. [3]
7. [Maximum mark: 6] 23M.1.SL.TZ2.9
The lengths of the seeds from a particular mango tree are approximated by a normal
distribution with a mean of 4 cm and a standard deviation of 0. 25 cm.

A seed from this mango tree is chosen at random.

(a) Calculate the probability that the length of the seed is less than 3. 7 cm. [2]

It is known that 30% of the seeds have a length greater than k cm.

(b) Find the value of k. [2]

For a seed of length d cm, chosen at random, P(4 − m < d < 4 + m) = 0. 6.

(c) Find the value of m. [2]

8. [Maximum mark: 5] 23M.1.SL.TZ2.12
In a game, balls are thrown to hit a target. The random variable X is the number of times the
target is hit in five attempts. The probability distribution for X is shown in the following table.

x 0 1 2 3 4 5

P(X = x) 0. 15 0. 2 k 0. 16 2k 0. 25

(a) Find the value of k. [2]

The player has a chance to win money based on how many times they hit the target.

The gain for the player, in $, is shown in the following table, where a negative gain means that
the player loses money.

x 0 1 2 3 4 5

Player’s gain ($) −4 −3 −1 0 1 4

(b) Determine whether this game is fair. Justify your answer. [3]
9. [Maximum mark: 6] 23M.1.AHL.TZ2.7
Akar starts a new job in Australia and needs to travel daily from Wollongong to Sydney and
back. He travels to work for 28 consecutive days and therefore makes 56 single journeys. Akar
makes all journeys by bus.

The probability that he is successful in getting a seat on the bus for any single journey is 0. 86.

(a) Determine the expected number of these 56 journeys for which Akar gets a
seat on the bus. [1]

(b) Find the probability that Akar gets a seat on at least 50 journeys during these
28 days. [3]

The probability that Akar gets a seat on at most n journeys is at least 0. 25.

(c) Find the smallest possible value of n. [2]

10. [Maximum mark: 5] 22N.1.SL.TZ0.8

Roy is a member of a motorsport club and regularly drives around the Port Campbell racetrack.

The times he takes to complete a lap are normally distributed with mean 59 seconds
and standard deviation 3 seconds.

(a) Find the probability that Roy completes a lap in less than 55 seconds. [2]

Roy will complete a 20 lap race. It is expected that 8. 6 of the laps will take more than t
seconds.

(b) Find the value of t. [3]

11. [Maximum mark: 7] 22N.1.SL.TZ0.9
Taizo plays a game where he throws one ball at two bottles that are sitting on a table. The
probability of knocking over bottles, in any given game, is shown in the following table.

(a) Taizo plays two games that are independent of each other. Find the
probability that Taizo knocks over a total of two bottles. [4]

In any given game, Taizo will win k points if he knocks over two bottles, win 4 points if
he knocks over one bottle and lose 8 points if no bottles are knocked over.

(b) Find the value of k such that the game is fair. [3]

12. [Maximum mark: 6] 22M.1.SL.TZ1.8

A factory produces bags of sugar with a labelled weight of 500 g. The weights of the bags are
normally distributed with a mean of 500 g and a standard deviation of 3 g.

(a) Write down the percentage of bags that weigh more than 500 g. [1]

A bag that weighs less than 495 g is rejected by the factory for being underweight.

(b) Find the probability that a randomly chosen bag is rejected for being
underweight. [2]

(c) A bag that weighs more than k grams is rejected by the factory for being
overweight. The factory rejects 2% of bags for being overweight.

Find the value of k. [3]

13. [Maximum mark: 5] 22M.1.SL.TZ2.10
The masses of Fuji apples are normally distributed with a mean of 163 g and a
standard deviation of 6. 83 g.

When Fuji apples are picked, they are classified as small, medium, large or extra
large depending on their mass. Large apples have a mass of between 172 g and 183 g.

(a) Determine the probability that a Fuji apple selected at random will be a large
apple. [2]

Approximately 68% of Fuji apples have a mass within the medium-sized category, which
is between k and 172 g.

(b) Find the value of k. [3]

14. [Maximum mark: 7] 22M.1.SL.TZ2.5

A polygraph test is used to determine whether people are telling the truth or not, but it is
not completely accurate. When a person tells the truth, they have a 20% chance of failing the
test. Each test outcome is independent of any previous test outcome.

10 people take a polygraph test and all 10 tell the truth.

(a) Calculate the expected number of people who will pass this polygraph test. [2]

(b) Calculate the probability that exactly 4 people will fail this polygraph test. [2]

(c) Determine the probability that fewer than 7 people will pass this polygraph
test. [3]
15. [Maximum mark: 7] 21M.1.SL.TZ1.10
A game is played where two unbiased dice are rolled and the score in the game is the greater
of the two numbers shown. If the two numbers are the same, then the score in the game is the
number shown on one of the dice. A diagram showing the possible outcomes is given below.

Let T be the random variable “the score in a game”.

(a) Complete the table to show the probability distribution of T .

[2]

Find the probability that

(b.i) a player scores at least 3 in a game. [1]

(b.ii) a player scores 6, given that they scored at least 3. [2]

(c) Find the expected score of a game. [2]

16. [Maximum mark: 6] 21M.1.AHL.TZ1.14
The weights of apples from Tony’s farm follow a normal distribution with mean 158 g
and standard deviation 13 g. The apples are sold in bags that contain six apples.

(a) Find the mean weight of a bag of apples. [2]

(b) Find the standard deviation of the weights of these bags of apples. [2]

(c) Find the probability that a bag selected at random weighs more than 1 kg. [2]

17. [Maximum mark: 6] 19N.1.SL.TZ0.T_12

The Malthouse Charity Run is a 5 kilometre race. The time taken for each runner to
complete the race was recorded. The data was found to be normally distributed with a mean
time of 28 minutes and a standard deviation of 5 minutes.

A runner who completed the race is chosen at random.

(a) Write down the probability that the runner completed the race in more than
28 minutes. [1]

(b) Calculate the probability that the runner completed the race in less than 26
minutes. [2]

(c) It is known that 20% of the runners took more than 28 minutes and less than
k minutes to complete the race.

Find the value of k. [3]

18. [Maximum mark: 6] 19M.1.SL.TZ1.T_11
Consider the following graphs of normal distributions.

(a) In the following table, write down the letter of the corresponding graph next
to the given mean and standard deviation.

[2]

At an airport, the weights of suitcases (in kg) were measured. The weights are normally
distributed with a mean of 20 kg and standard deviation of 3.5 kg.

(b) Find the probability that a suitcase weighs less than 15 kg. [2]

(c) Any suitcase that weighs more than k kg is identified as excess baggage.
19.6 % of the suitcases at this airport are identified as excess baggage.

Find the value of k. [2]

19. [Maximum mark: 13] 19M.1.SL.TZ1.S_9
A random variable Z is normally distributed with mean 0 and standard deviation 1. It is
known that P(z < −1.6) = a and P(z > 2.4) = b. This is shown in the following diagram.

(a) Find P(−1.6 < z < 2.4). Write your answer in terms of a and b. [2]

(b) Given that z > −1.6, find the probability that z < 2.4 . Write your answer in
terms of a and b. [4]

A second random variable X is normally distributed with mean m and standard deviation s.

It is known that P(x < 1) = a.

(c) Write down the standardized value for x = 1. [1]

(d) It is also known that P(x > 2) = b.

Find s. [6]
20. [Maximum mark: 6] 19M.1.SL.TZ2.T_14
The price per kilogram of tomatoes, in euro, sold in various markets in a city is found to be
normally distributed with a mean of 3.22 and a standard deviation of 0.84.

(a.i) On the following diagram, shade the region representing the probability that
the price of a kilogram of tomatoes, chosen at random, will be higher than
3.22 euro.

[1]

(a.ii) Find the price that is two standard deviations above the mean price. [1]

(b) Find the probability that the price of a kilogram of tomatoes, chosen at
random, will be between 2.00 and 3.00 euro. [2]

(c) To stimulate reasonable pricing, the city offers a free permit to the sellers
whose price of a kilogram of tomatoes is in the lowest 20 %.

Find the highest price that a seller can charge and still receive a free permit. [2]

21. [Maximum mark: 7] 19M.1.AHL.TZ1.H_6

Let X be a random variable which follows a normal distribution with mean μ. Given that
P (X < μ − 5) = 0.2 , find

(a) P (X > μ + 5). [2]

(b) P (X < μ + 5 | X > μ − 5). [5]

22. [Maximum mark: 15] 18N.1.SL.TZ0.S_9
A bag contains n marbles, two of which are blue. Hayley plays a game in which she randomly
draws marbles out of the bag, one after another, without replacement. The game ends when
Hayley draws a blue marble.

(a.i) Find the probability, in terms of n, that the game will end on her first draw. [1]

(a.ii) Find the probability, in terms of n, that the game will end on her second draw. [3]

Let n = 5. Find the probability that the game will end on her

(b.i) third draw. [2]

(b.ii) fourth draw. [2]

(c) Hayley plays the game when n = 5. She pays $20 to play and can earn money
back depending on the number of draws it takes to obtain a blue marble. She
earns no money back if she obtains a blue marble on her first draw. Let M be
the amount of money that she earns back playing the game. This information
is shown in the following table.

Find the value of k so that this is a fair game. [7]

23. [Maximum mark: 6] 18N.1.SL.TZ0.T_14
The marks achieved by students taking a college entrance test follow a normal distribution
with mean 300 and standard deviation 100.

In this test, 10 % of the students achieved a mark greater than k.

(a) Find the value of k. [2]

Marron College accepts only those students who achieve a mark of at least 450 on the test.

(b) Find the probability that a randomly chosen student will be accepted by
Marron College. [2]

(c) Given that Naomi attends Marron College, find the probability that she
achieved a mark of at least 500 on the test. [2]

24. [Maximum mark: 6] 18M.1.SL.TZ1.T_13

Malthouse school opens at 08:00 every morning.

The daily arrival times of the 500 students at Malthouse school follow a normal distribution.
The mean arrival time is 52 minutes after the school opens and the standard deviation is 5
minutes.

(a.i) Find the probability that a student, chosen at random arrives at least 60
minutes after the school opens. [2]

(a.ii) Find the probability that a student, chosen at random arrives between 45
minutes and 55 minutes after the school opens. [2]

(b) A second school, Mulberry Park, also opens at 08:00 every morning. The arrival
times of the students at this school follows exactly the same distribution as
Malthouse school.

Given that, on one morning, 15 students arrive at least 60 minutes after the
school opens, estimate the number of students at Mulberry Park school. [2]
25. [Maximum mark: 5] 18M.1.AHL.TZ1.H_3
Two unbiased tetrahedral (four-sided) dice with faces labelled 1, 2, 3, 4 are thrown and the
scores recorded. Let the random variable T be the maximum of these two scores.

The probability distribution of T is given in the following table.

(a) Find the value of a and the value of b. [3]

(b) Find the expected value of T. [2]

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