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CHAPTER 8 : PROBABILITY DISTRIBUTIONS
1. Understand and use the concept of binomial distribution.
1.1 List all possible values of a discrete random variable.
(a) If Xrepresents the number of pupilsscoring 12A in a group of 5 pupils, list
all the possible values ofX.
(b) If Yrepresents the number of times ofgetting the number 1 when tossing a
fair dice three times, list all thepossible values ofY.
(c) A pupil takes English examination 8times. IfZrepresents the number of
times he passes the examination, list
all the possible values ofZ.
(d) 3 marbles are chosen from a bagcontaining 5 red marbles and 4 black
marbles. IfXrepresents the number
of black marbles chosen, list all thepossible values ofX.
1.2 Determine the probability of an event in a binomial distribution. ( ) , 1
n r n r
rP X r C p q p q
= = + =
(a) Givenp = 0.7 , q =
n = 5 , r=3Find P(X= 3).
[0.3087]
(b) Given p = 0.2 , q =
n = 10 , r=2Find P(X= 2).
[ 0.3020]
(c) Given p = 0.45 , q =
n = 7 , r=3
Find P(X= 3).
[0.2918]
(d) Given p = 0.12 , q =
n = 10 , r=2
Find P(X= 2).
[ 0.2330]
(e) The probability that Bernard will be
late for a meeting is 0.6. Find the
probability that Bernard will be latefor 3 out of five meetings.
[ 0.3456]
(f) The probability that Minmin scoring
1A for English in the monthly test is
0.4. Find the probability that Minminwill be scoring 1A for English twice
out of 6 monthly tests.
[ 0.3110]
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(g
)
The probability that Chandra wins in
a singing competition is 0.65. Findthe probability that Chandra will win
4 out of 6 singing competitions.
[ 0.3280]
(h) The probability that Ahmad wears
batik shirt for a meeting is 0.55. Findthe probability that Ahmad will wear
batik shirt for 2 out of five meetings.
[ 0.2757]
1.3 Plot binomial distribution graphs.
(a) Siva has to play 4 games. The
probability that Siva will win a game
is 0.55. If the binomial randomvariableXrepresents the number of
games that Siva won,
find the probability that Siva wins 0,
1, 2, 3 or 4 games .Plot a binomial distribution graph.
A dice is tossed 4 times. The
probability of getting the number 2
is9
20. If the binomial random
variableXrepresents the number of
times of getting the number 2,find the probability of getting
0,1,2,3,or 4 times of number 2
Plot a binomial distribution graph.
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1.4 Determine mean, variance and standard deviation of a binomial distribution.
2
; ; ; 1np npq npq p q = = = + =
(a)Given n =200,
1
6p = , Find the value
of
i) q, ii)
iii) 2 iv)
(b)Given n =100,
1
3p = , Find the value
of
i) q, ii)
iii) 2 iv)
(c) Johnny attempted 60 questions with 4
options to choose from. There is onlyone correct answer for each question.
Johnny guessed all the answers .
Find(a) the mean number of questions that
he will get it right.
(b) the variance and the standarddeviation of the number of correctanswers obtained.
[15, 11.25 ,3.354]
(d) Given that a class consists of 30
students, 70% of them pass in amathematics test. Find
(a) the mean number of students pass
the test.(b) the variance and the standard
deviation of the number of
students pass the test
[21, 6.3, 2.510]
(e) A fair dice is rolled 15 times
continuously. The probability ofobtaining the number 2 is
1
6. Find
(a) the mean number of times that
number 2 appears
(b) the variance and the standarddeviation of number 2 is
obtained.
[2.5, 2.083,1.443]
(f) Given that 30 bombs were released
by a jet fighter, the probability that abomb will hit its target is 0.65. Find
(a) the mean number bombs hitits target.
(b) the variance and the standard
deviation of the number of bombshit its target
[19.5,6.825,2.612]
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1.5 Solve problems involving binomial distributions. ( ) , 1n r n r
rP X r C p q p q
= = + =
(a) In SMK Simpang Lima, theprobability of a pupil being latein
Form 3D is 8%. Calculate the
probability that in a group of 10
students from Form 3D, the numberof pupils being late is
(i) exactly two
(ii) less than two
[ 0.1478, 0.8121]
(b) Given that 12 bombs were releasedby a jet fighter, the probability that a
bomb will hit its target is 0.65.
calculate the probability that
(a) exactly 10 bombs hit the target(b) all the bombs hit the target
[ 0.1088, 0.005688]
(c) In Kuantan, the probability that a
teenager owns a mobile phone is 45
.
A group of 10 teenagers are selected
at random from Kuantan. Find theprobability
(i) that exactly 7 of the selected
teenagers own a mobile phone.
(ii) at least 8 of the selectedteenagers own a mobile phone.
[ 0.2013, 0.6778]
(d) 72 % of the pupils in Form 5D pass
their aptitude test. If 4 pupils areselected at random from the class.
Calculate the probability that
(i) at least half of the selected pupilspass their aptitude test
(ii) at most 1 of the selected pupils
pass heir aptitude test,
[ 0.9306, 0.0694]
(e) In a shooting competition, the chance
for John to hit the target on any oneshot is 95%. John fires 8 shots. Find
the probability that
(i) at least 7 shots hit the target(ii) at most 3 of the shots hit the
target.
[0.9428, 0.0000154]
(f) A fair dice is tossed 10 times
continuously. The probability of
obtaining the number 2 is1
6. Find
the probability of getting
(i) number 2 not more than 3
tosses
(ii) number 2 in 4 or more tosses
[ 0.9303,0.0697]
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(g) In an examination, 70% of the studentspassed. If a sample of 8 students is
randomly selected, find the probability
that 6 students from the sample passed
the examination.
[0.2965]
(h) Senior citizens make up 20% of thepopulation of a settlement.
If 7 people are randomly selected from
the settlement, find the probability that at
least two of them are senior citizens.
[0.4233]
(i) The result of a study shows that 20% of
the pupils in a city cycle to school. If 8pupils from the city are chosen at
random, calculate the probability that
(i) exactly 2 of them cycle to school
(ii) less than 3 of them cycle to school
[0.2936, 0.79691]
(j) The probability that each shot fired by
Ramli hits a target is 1/3(i) If Ramli fires 10 shots, find the
probability that exactly 2 shots hit
the target.
(ii) If Ramli fires n shots, the
probability that all the n shots hit
the target is 1/243. Find the value
of n.
[0.1951, 5]
(k) It is known that 18 out of 30 students in aclass like to read during their free time. 9
students are selected at random from the
class. Find the probability thatall the selected students like to read
during their free time.
(ii) at least 7 of the selected students like
to read during their free time.
[ 0.01008, 0.2318]
(l) In an IT literacy research in a village, itis found that every one out of three
homes has computers. If 10 homes are
randomly selected, find the probabilitythat
(i) not a single home has computers
(ii) at least two homes have computers.
[0.01734, 0.8960]
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2. Understand and use the concept of normal distribution.
2.1 Describe continuous random variables using set notations.
2.2 Find probability ofz-values for standard normal distribution.
1 Given thatZis the standard normal distribution variable, find the values for the
following:
(a) P(Z>0.2)
[ 0.4207 ]
(b) P(Z>1.2)
[ 0.1151 ]
(c) P(Z< -0.6 )
[ 0.2743]
(d) P(Z< -1.5)
[ 0.0668]
(e) P( Z > -1.511)
[0.9346]
(f) P(Z> -0.203)
[0.5805]
(g) P(Z< 1.327)
[0.9077]
(h) P(Z< 0.549 )
[0.7085]
(i) P( 0.2 < Z< 1.2)
[0.3056]
(j) P( 0.548
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(o) P( 1.334Z )
[0.1824]
(p) P( 0.625Z )
[ 0.5320](q) P( 1.112Z )
[ 0.7338]
(r) P( 0.336Z )
[ 0.2630]
2 Find thez-score for each of the following:
(a) P( Z > z) =0.4207
[0.2]
(b) P(Z > z) =0.1151
[1.2]
(c) P( Z < z) =0.2743
[-0.6]
(d) P( Z < z) =0.0668
[-1.5]
(e) P( Z > z) = 0.5805
[-0.203 ]
(f) P( Z > z) = 0.9346
[-1.511]
(g) P( Z < z) = 0.9332
[1.5]
(h) P( Z< z) = 0.8757
[1.154]
(i) P( -1.2 Z z ) 0.2369, z
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2.3 Convert random variable of normal distributions,X, to standardised variable,Z.
(1) For each normal distribution below, convert the random variable X to standardized variable
Z. Then, find the probability of the event given the mean and the standard deviation.
(a)
X 12 10 3
XZ
=
Probabilit
y of event
P(Z>z)
[ 0.667, 0.2523]
(b)
X 43.26 50 10
XZ
=
Probabilit
y of event
P(Zz)
[ -0.2 , 0.5793]
(d) X 43 50 8
XZ
=
Probabilit
y of event
P( Z >z)
[ -0.875, 0.6184]
2.4 Represent probability of an event using set notation.
2.5 Determine probability of an event.
2.6 Solve problems involving normal distributions.
(a) Xis random variable of a normal
distribution with mean and standard
deviation 6. Find
(i) the value of , if thez-value is 1.5
whenX= 42
(ii)P(X
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(c) Xis random variable of a normal
distribution with mean 6.7 and standard
deviation 1.5. Find
(i) thez-score givenX=4.6
(ii)P(X 4.6)
[-1.4, 0.9192]
(d) Xis random variable of a normal
distribution with mean and variance
2.25. Find
(i) the value of if thezscore =2
whenX= 8.5(ii) P(X< 7)
[5.5, 0.8413]
(e) Xis random variable of a normal
distribution with mean 108 and standard
deviation 10. Find(i) the Zscore ifX=100
(ii)P( 100 X 108)
[-0.8, 0.2881]
(f) Xis random variable of a normal
distribution with mean 24 and standard
deviation 2.5. Find(i)P(X>26 )
(ii) value ofkifP(X< k) = 0.1151
[ 0.2119, 27]
(g) Xis random variable of a normal
distribution with = 10 and standard
deviation 3. . Find the value ofk if
P(X< k) = 0.975
[15.88]
(h) Xis a continuous random variable such
that X~N( )25, . Given that
P(X < 9)= 0.7257. Find the value of .
[6]
(i) A random variableXhas a normal
distribution with mean 50 and variance,2
. Given that
P(X > 51) = 0.288, find the value of
[1.795]
(j) Xis random variable of a normal
distribution with mean 38.5 and
variance 10.24 . Find the value of a
givenP(X > a) = 0.268
[44.84]
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(k) Diagram below shows a standard normal
distribution graph.
IfP(0 k)
[0.1872]
(l) The diameters of the marbles produced
by a factory are normally distributed
with a mean of 9mm and a standard
deviation of 0.1 mm. Diagram below
shows the normal distribution graph for
the diameter of the marbles,Xmm
It is given that the area of the shaded
region is o.4522. Find the value of h
[9.012]
(m) Diagram below shows a standardized
normal distribution graph.
The probability represented by the area
of the shaded region is 0.3643(i) Find the value ofk.
(ii) X is a continuous random variable
which is normally distributed with a
mean of and a standard deviation of
8.
Find the value of ifX= 70 when thez-
score is k.
[ 1.1, 61.2]
(n)
Diagram shows a standard normal
distribution graph.
The probability represented by the area
of the shaded region is 0.3485.
(i)Find the value of k
(ii)Xis a continuous random variablewhich is normally distributed with a
mean of 79 and a standard deviation of
3. Find the value ofXwhen thez-score is
k
[ 1.03, 82.09]
8. Probability Distributions 10
F(z)
k00=9
Xmmh
0.3643
zkO
f(z)
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(o) Marks obtained by a group of students in
an examination are normally distributed
with a mean of 48 marks and standard
deviation of 10 marks.
(i) If a student is selected at random,
calculate the probability that the students
mark is more than 60(ii)If 30% of the students fail in the
examination, estimate the minimum mark
required to pass the examination.
[0.1151,42.76]
(p) A group of workers are given medical
check up. The blood pressure of a
worker has a normal distribution with a
mean of 130 mmHg and a standard
deviation of 16mmHg. Blood pressure
that is more than 150 mmHg is classify
as high blood pressure(i) a worker is chosen at random from
the group. Find the probability that the
worker has a blood pressure between
114mm Hg and 150mmHg(ii) It is found that 132 workers have
high blood pressure. Find the total
number of workers in the group.
[ 0.7357, 1249]
(q) The mass of a packet of cookies has a
normal distribution with mass of 248 g
and a standard deviation of 16 g. Find
(i) The probability that a packet of
cookies is chosen at random has massbetween 240 g and 250 g.
(ii) the value ofx if 20% of the packet of
cookies chosen at random have a mass
greater thanx g
[0.2412, 261.47]
(r) In a training session with a group of
Form 5 boys, it is discovered that the
rate of heart beats has a normal
distribution with a mean of 80 beats per
minute and a standard deviation of 10beats per minutes.
(i) If one form 5 boy is randomly
selected, find the probability that
his rate of heart beats is between75 to 85 beats per minutes.
(ii) Given that 20% of the Form 5
boys have rate of heart bets less
than kbeats per minute, find the
value ofk.
[0.383, 71.58]
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(s) The life span of a type of battery
produced by a factory is normally
distributed with mean 325 hours and
standard deviation of 25 hours. Find
(i) the probability that a unit of battery
chosen at random, has a life span
between 280 hours and 350 hours,(ii) the percentage of battery that has a
life span of more than 320 hours.
[0.8054, 57.93%]
(t) The mass of papayas produced by an
orchard is normally distributed with a
mean and variance 0.64 g
(i) Given that 10.5% of the papayas
produced exceed 3.5 kg. Find the value
of
(ii) The papayas that have masses lessthan 1.0 kg and more than 4.0 kg are
rejected from packaging. Calculate the
percentage of acceptable papayas.
[ 2.4976, 93.92%]
(u) The mass of water melons produced from
an orchard follows a normal distribution
with a mean of 3.2kg and a standard
deviation of 0.5 kg. find
(i) the probability that a water-melon
chosen randomly from the orchard have a
mass of not more than 4.0 kg(ii) the value of m if 60% of the water
melons from the orchard have a mass
more than m kg
[ 0.9452,3.0735,]
(v) The mass of mango fruits from a farm is
normally distributed with a mean of
820 g and standard deviation of 100 g.
(i)Find the probability that a mango fruit
chosen randomly has a minimum mass
of 700 g.
(ii)Find the expected number of mangofruits from a basket containing 200 fruits
that have a mass of less than 700 g.
[ 0.8849, 23 ]
w (a)Government servants make up 15% of x (a) A football team is having a practice
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the population of a village.
(i) If 8 people are selected randomly from
the village, find the probability that at
least two of them are government
servants.
(ii) If the variance of government
servants is 91.8, what is the population ofthe village? [5 marks]
(b) The height of the workers in a factory
is normally distributed with a mean of
162.5 cm and a variance of 90.25 cm2.180 of the workers are between 151.3 cm
and 169.7 cm tall. Find the total number
of workers in the factory. [ 5marks]
[0.3428, 720,268]
session on scoring goals from penalty
kicks. Each player takes 10 penalty . The
probability that a player scores a goal
from a penalty kick isp. After the
practice session, it is found that the mean
number of goals for a player is 3.6
(i) Find the value ofp.(ii) If a player is chosen at random, find
the probability that he scores at least one
goal. [5 marks]
(b) The heights of students are normallydistributed with a mean of 162 cm, and a
standard deviation of 12 cm
(i) If a student is chosen at random,
calculate the probability that his height
is less than 154 cm.
(ii)Given that 15% of the students are
taller than h cm, find the value ofh
[5 marks][0.36, 0.9885, 0.2523,174.4]
2.7 Past year questions
(a) SPM 2008 P2 Q11The masses of mangoes from an orchard
has a normal distribution with a mean of
300 g and a standard deviation of 80 g.
(a) Find the probability that a mango
chosen randomly from this orchard has a
mass of more than 168 g [3marks](b) A random sample of 500 mangoes is
chosen.
(i) calculate the number of mangoes from
this sample that have a mass of more than
168 g
(ii) Given that 435 mangoes from this
sample have a mass of more than m g,
find the value of m [7marks]
[0.95053,475/476, 209.84]
(b) SPM 2007 P2 Q11(a) In a survey carried out in a school, it
is found that 2 out of 5 students have
handphones. If 8 students from that
school are chosen at random, calculate
the probability that
(i) exactly 2 students have handphones(ii) more than 2 students have
handphones [5marks]
(b) A group of workers are given
medical check up. The blood pressure of
a worker has a normal distribution with a
mean of 130 mmHg and a standard
deviation of 16mmHg. Blood pressure
that is more than 150 mmHg is classify
as high blood pressure
(i) a worker is chosen at random from
the group. Find the probability that the
worker has a blood pressure between
114mm Hg and 150mmHg(ii) It is found that 132 workers have
high blood pressure. Find the total
number of workers in the group.
[5marks]
[0.6846, 0.7357, 0.1056,1250]
(c) SPM 2006 P2 Q 11 (d) SPM 2005 P2 Q11
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An orchard produces lemons. Only
lemons with diameter, x greater than k
cm are graded and marketed. Table
below show the grades of the lemons
based on their diameters.
Grade A B C
Diameter,x(cm)
X > 7 7 > x > 5 5 > x > k
It is given that the diameter of the lemons
has a normal distribution with a mean of
5.8 cm and a standard deviation of 1.5
cm
(a) If one lemon is picked at random,
calculate the probability that it is of grade
A [2marks](b) In a basket of 500 lemons, estimate
the number of grade B lemons.
[4marks]
(c) If 85.7% of the lemons is marketed,find the value of k.
[4marks]
[ 0.2119, 0.4912, 4.1965]
(a) The result of a study shows that 20%
of the pupils in a city cycle to school. If
8 pupils from the city are chosen at
random, calculate the probability that
(i) exactly 2 of them cycle to school
(ii) less than 3 of them cycle to school
[4marks](b) The mass of water melons produced
from an orchard follows a normal
distribution with a mean of 3.2kg and a
standard deviation of 0.5 kg. find(i) the probability that a water-melon
chosen randomly from the orchard have
a mass of not more than 4.0 kg
(ii) the value of m if 60% of the water
melons from the orchard have a mass
more than m kg [6marks]
[ 0.2936, 0.79691, 0.9452,3.0735,]
(e) SPM 2004 P2 Q11
(a) A club organizes a practice session
for trainees on scoring goals from penalty
kicks.. each trainee takes 8 penalty kicks.
The probability that a trainee scores a
goal from a penalty kick isp. After thesession, it is found that the mean number
of goals for a trainee is 4.8
(i) Find the value ofp.
(ii) If a trainee is chosen at random, findthe probability that he scores at least one
goal. [5 marks]
(b) A survey on body-mass is done on a
group of students. The mass of a student
has a normal distribution with a mean of
50kg and a standard deviation of 15 kg
(i) If a student is chosen at random,
calculate the probability that his mass isless than 41kg
(ii) Given that 12% of the students have a
mass of more than m kg, find the value of
m. [5 marks]
[0.6, 0.9993,0.2743,67.63]
(f) SPM 2003 P2 Q 10
(a) Senior citizens make up 20% of the
population of a settlement.
(i) If 7 people are randomly selected
from the settlement, find the probability
that at least two of them are seniorcitizens
(ii) If the variance of the senior citizens
in 128, what is the population of the
settlement. [5 marks](b) The mass of the workers in a factory
is normally distributed with a mean of
67.86 kg and a variance of 42.25 kg2.
200 of the workers in the factory weigh
between 50 kg and 70kg.
Find the total number of worker in the
factory. [5 marks]
[0.2, 0.4232, 800, 0.6259]
8. Probability Distributions 14