8. Probability Distributions

7/30/2019 8. Probability Distributions

1/14

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]

8. Probability Distributions 1


2/14

(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.



3/14

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]



4/14

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]



5/14

(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]



6/14

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


7/14

(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


8/14

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


9/14

(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]



10/14

(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]


F(z)

k00=9

Xmmh

0.3643

zkO

f(z)


11/14

(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]



12/14

(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



13/14

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



14/14

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


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


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

Documents

Transcript of 8. Probability Distributions