Business Mathematics and Statistics (MATH0203)

Probability

Ms Nurazrin Jupri

Probability is the chance that something will happen. It is a measure of likelihood and can be stated as a percentage, a ratio, or more usually

as a number from 0 to 1.

Consider the following. • Probability = 0 = impossibility • Probability = 1 = certainty • Probability = 1/2 = a 50% chance of something happening • Probability = 1/4 = a 1 in 4 chance of something happening

Ms Nurazrin Jupri

What is probability?

• A probability is the chance that something will

happen.

• Eg:

▫ (a) I am 100% confidence that I will pass the exam.

▫ (b) About 50% of the staff are not satisfied with their salary.

▫ (c) The chance for you to win this battle is very low.

Ms Nurazrin Jupri

Experiment

• An experiment is a situation involving chance

or probability that leads to results called outcomes.

• Eg:

▫ 1. Rolling a fair die.

▫ 2. Tossing a coin.

▫ 3. Choosing a student from the class.

Ms Nurazrin Jupri

Outcomes

• An outcome is the result of a single trial of an

experiment.

• Eg:

▫ 1. Getting a ‘1’ in a rolling of a fair die.

▫ 2. Getting a ‘head’ in a tossing of a coin.

Ms Nurazrin Jupri

Sample Space, (S)

• is the set of all possible outcomes of an

experiment.

• Eg:

▫ 1. All the outcomes for the experiment ‘Rolling a fair die’: 𝑆 = 1,2,3,4,5,6

▫ 2. All the outcomes for the experiment ‘Tossing a coin’: 𝑆 = 𝐻, 𝑇

▫ 3. All the outcomes for the experiment ‘Choosing a student from the class’: _____________

Ms Nurazrin Jupri

Example 1

• List the sample space for the following experiment. An experiment of

▫ Tossing two coins

▫ Tossing three coins

For each of the experiment above, show the presentation of outcomes in table and tree diagram

Ms Nurazrin Jupri

• Tossing two coins

By table

By tree diagram

H ……… HH

H T ……… HT

T H ……… TH

T ……… TT

Coin 1 H H T T

Coin 2 H T H T

𝑆 = 𝐻𝐻, 𝐻𝑇, 𝑇𝐻, 𝑇𝑇

Ms Nurazrin Jupri

• Tossing three coins

By table

Coin 1

Coin 2

Coin 3

𝑆 = 𝐻𝐻𝐻, 𝐻𝐻𝑇, 𝐻𝑇𝐻, 𝐻𝑇𝑇, 𝑇𝐻𝐻, 𝑇𝐻𝑇, 𝑇𝑇𝐻, 𝑇𝑇𝑇

Coin 1 H H H H T T T T

Coin 2 H H T T H H T T

Coin 3 H T H T H T H T

Ms Nurazrin Jupri

By tree diagram

H ……… HHH

H T ……… HHT

H T H ……… HTH

T ……… HTT

T H H ……… THH

T T ……… THT

H ……… TTH

T ……… TTT

𝑆 = 𝐻𝐻𝐻, 𝐻𝐻𝑇, 𝐻𝑇𝐻, 𝐻𝑇𝑇, 𝑇𝐻𝐻, 𝑇𝐻𝑇, 𝑇𝑇𝐻, 𝑇𝑇𝑇

Ms Nurazrin Jupri

Event

• An event is one or more outcomes that may

occur. • Is a subset of sample space. • Eg:

▫ 1. Getting even numbers for the experiment ‘Rolling a fair die’. E = 2,4,6

▫ 2. Getting multiplies of 3 for the experiment ‘Rolling a fair die’. M3 = 3,6

▫ 3. Getting long hair students for the experiment ‘Choosing a student from the class’._______

Ms Nurazrin Jupri

Complement

• The complement of an event A with respect to S is the subset of all elements of S that are not in A.

• The complement of A denoted by symbol A’

Venn diagram

s

𝑆 = 𝐴 ∪ 𝐴′

Ms Nurazrin Jupri

Example 2

• Let 𝑆 = 𝐴𝑙𝑔𝑒𝑏𝑟𝑎, 𝐶𝑎𝑙𝑐𝑢𝑙𝑢𝑠, 𝑆𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐𝑠, 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛𝑠 and A = 𝐴𝑙𝑔𝑒𝑏𝑟𝑎, 𝐶𝑎𝑙𝑐𝑢𝑙𝑢𝑠 . Find A’

𝐴′ = 𝑆𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐𝑠, 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛𝑠

Ms Nurazrin Jupri

Intersection

• The intersection of two events A and B, denoted by symbol 𝐴 ∩ 𝐵.

• The event containing all elements that are common to A and B

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Example 3

• Given 𝑆 = 1, 2,3,4,5,6,7,8,9,10 , A = 2,4,6,8,10 and B = 2,3,5,7 . Find 𝐴 ∩ 𝐵

𝐴 ∩ 𝐵 = 2

.3 .6

.10 .8

.4

.2 .7

.5

.9

.1 S

Ms Nurazrin Jupri

Mutually exclusive

• Two events A and B are mutually exclusive, if 𝐴 ∩ 𝐵 ≠ ∅

• A and B have no elements that are common

• (a) I am female and I am male. – Impossible!

(b) I am 6 years old and I am 25 years old. – Impossible!

Ms Nurazrin Jupri

Example 4

• Given 𝑆 = 1, 2,3,4,5,6,7,8 , A = 2,4,6 and B = 1,3,5 . Are A and B mutually exclusive?

𝐴 ∩ 𝐵 = ∅

.3 .6

.8

.4

.2 .7 .5

.1

S

Ms Nurazrin Jupri

Union

• The union of the two events A and B, denoted by the symbol 𝐴 ∪ 𝐵

• Is the event containing all the elements belong to A or B or both.

S

𝐴 ∪ 𝐵

Ms Nurazrin Jupri

Example 5

• Given 𝑆 = 1, 2,3,4,5,6,7,8,9,10 , A = 2,4,6,8,10 and B = 2,3,5,7 . Find 𝐴 ∪ 𝐵 and n 𝐴 ∪ 𝐵 .

𝐴 ∪ 𝐵 = 2,3,4,5,6,7,8,10 .3 .6

.10 .8

.4

.2 .7

.5

.9

.1 S

𝑛(𝐴 ∪ 𝐵) = 8

Ms Nurazrin Jupri

Probability of an Event

𝑃 𝐴 =𝑛(𝐴)

𝑛(𝑆)

Where:

• n(A) = the number of outcomes in event set A

• n(S) = sample space (All possible outcome)

Ms Nurazrin Jupri

Properties of Probability

1. 0 ≤ 𝑃 𝐴 ≤ 1

2. 𝑃 𝑆 = 1

3. 𝑃 ∅ = 0

Ms Nurazrin Jupri

Example 6

• A pair of coins is tossed. Find the probability of getting

a) Two heads

b) One tail and one head

c) No head

Let H=head, T=tail 𝑆 = 𝐻𝐻, 𝐻𝑇, 𝑇𝐻, 𝑇𝑇 and n 𝑆 = 4

a) Let A = 𝐻𝐻 and n 𝐴 = 1, thus 𝑃 𝐴 = 1

4

b) Let B = 𝐻𝑇, 𝑇𝐻 and n 𝐵 = 2, thus 𝑃 𝐵 = 2

4=

1

2

c) Let C = 𝑇𝑇 and n 𝐶 = 1, thus 𝑃 𝐶 = 1

4

Ms Nurazrin Jupri

Example 7

• A mixture of candies contains 6 mints, 4 toffees and 3 chocolates. If a person makes a random selection of one of these candies, find the probability of getting

a) A mint

b) A toffee or a chocolate

Let H=mint, T=toffee, C=chocolate; n 𝑆 =13

a) Since there are 6 mint candies n 𝑀 = 6, thus 𝑃 𝑀 = 6

13

b) Since there are 7 candies are toffee or chocolate n 𝑇 ∪ 𝐶 = 7, thus

𝑃 𝑇 ∪ 𝐶 = 7

13

Ms Nurazrin Jupri

Example 8

• A box contains 4 yellow balls and 6 black balls. Two balls were taken from the box, with replacing the first ball before taking second ball. Find

a) 𝑃(2 𝑦𝑒𝑙𝑙𝑜𝑤 𝑏𝑎𝑙𝑙𝑠)

b) 𝑃(𝑡𝑤𝑜 𝑏𝑎𝑙𝑙𝑠 𝑤𝑖𝑡ℎ 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑐𝑜𝑙𝑜𝑟𝑠)

c) 𝑃(𝑎𝑡 𝑙𝑒𝑎𝑠𝑡 𝑜𝑛𝑒 𝑦𝑒𝑙𝑙𝑜𝑤 𝑏𝑎𝑙𝑙)

Let Y=yellow, B=black; 𝑆 = 𝑌𝑌, 𝑌𝐵, 𝐵𝑌, 𝐵𝐵 and n 𝑆 = 4 a) P 2 yellow balls = P YY

=4

10×

4

10=

4

25

a) P 2 balls 𝑤𝑖𝑡ℎ 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑐𝑜𝑙𝑜𝑟𝑠 = P Y𝐵 ∪ 𝐵𝑌

=4

10×

6

10+

6

10×

4

10=

12

25

a) P at least one yellow ball = P Y𝐵 ∪ 𝐵𝑌 ∪ 𝑌𝑌

=4

10×

6

10+

6

10×

4

10+

4

10×

4

10=

16

25

Ms Nurazrin Jupri

• If A and A’ are complementary events, then 𝑃 𝐴 + 𝑃 𝐴′ = 1

Complementary Events

Example 9 • The probability that Pahang will win the badminton

tournament is 5/7. Find the probability that Pahang will lose?

Given 𝑃 𝑤𝑖𝑛𝑠 =5

7, thus 𝑃 𝑙𝑜𝑠𝑒𝑠 = 1 −

5

7=

2

7

Ms Nurazrin Jupri

Exercise 1

1. What is probability of getting a number 4 with the rolling of

a single die? Hence find the probability of not getting a

number 4.

2. The table shows the number of students in different forms

in a school. A student is selected from the school. Find the

probability that

▫ a Form 2 student is selected.

▫ a Form 5 student is selected.

Form 1 2 3 4 5

No. of students 40 80 70 60 50

Ms Nurazrin Jupri

Exercise 1 (cont.)

3. The test marks for mathematics in a class are recorded as

shown below:

▫ If a student in the class is at random, what is the probability that

he or she is getting 70 marks?

▫ Find the probability of a student, selected at random, scoring 90

marks and above.

Test marks 40 50 60 70 80 90 100

No. of students 4 8 10 12 8 9 2

Ms Nurazrin Jupri

• For two events A and B, 𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃(𝐴 ∩ 𝐵)

• For three events A, B and C, 𝑃 𝐴 ∪ 𝐵 ∪ 𝐶

= 𝑃 𝐴 + 𝑃 𝐵 + 𝑃 𝐶 − 𝑃 𝐴 ∩ 𝐵− 𝑃 𝐴 ∩ 𝐶 − 𝑃 𝐵 ∩ 𝐶 + 𝑃(𝐴 ∩ 𝐵 ∩ 𝐶)

Probability of Events

Additive Rule

Note: means OR! means AND!

Ms Nurazrin Jupri

Example 10

• The probability of Sarah passes Mathematics is 2

3, and

the probability that she passes English is 8

9. If the

probability of passing both subjects is 3

4, what is the

probability that Sarah will pass at least one of these subjects?

Let M=pass in Mathematics, E=pass in English; 𝑃 𝑀 ∪ 𝐸 = 𝑃 𝑀 + 𝑃 𝐸 − 𝑃 𝑀 ∩ 𝐸

=2

3+

8

9−

3

4=

29

36

Ms Nurazrin Jupri

• For two events A and B, 𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃 𝐴 ∩ 𝐵

𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 0 𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵

Additive Rule for Mutually Exclusive

Events

Note: means OR! means AND!

Ms Nurazrin Jupri

Example 11

• Four friends, Adam, Badri, Haziq and Rizal apply for a job in a petrol station but only one person will get the job. Find probability that Badri or Haziq will get the job.

Let B=Badri, H=Haziq; 𝑃 𝐵 ∪ 𝐻 = 𝑃 𝐵 + 𝑃 𝐻 − 𝑃 𝑀 ∩ 𝐸

=1

4+

1

4− 0 =

1

2

Ms Nurazrin Jupri

Exercise 2

1. John and Owen applied for a scholarship and only one student will be

awarded the scholarship. If the probability for John to be awarded the

scholarship is 0.3 and that for Owen is 0.4, what is the probability that,

either John or Owen to win the scholarship?

2. In the senior year of high school graduating class of 100 students, 42

studied Mathematics, 68 studied Psychology, 54 studied History, 22

studied both Mathematics and History, 25 studied both Mathematics and

Psychology, 7 studied History but neither mathematics nor Psychology,

10 studied all three subjects and 8 did not take any of the three.

Draw the Venn diagram

If a student is selected at random, find the probability that a person not taking

Psychology is taking both History and Mathematics.

Ms Nurazrin Jupri

Exercise 2 (cont,)

3. Find the probability of drawing a single card

with either a ‘7’ or a ‘heart’ from a pack of 52

playing cards.

Ms Nurazrin Jupri

• The probability that an event will occur, given that one or more other events have occurred.

• The conditional probability of B, given A,

denoted by 𝑃 𝐵 𝐴 =𝑃(𝐵∩𝐴)

𝑃(𝐴)

Conditional Probability

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Example 12

• The probability that a regularly schedules flight departs on time is 𝑃 𝐷 = 0.83, it arrives on time is 𝑃 𝐴 = 0.82 and it departs and arrives on time is 𝑃 𝐷 ∩ 𝐴 = 0.78. Find the probability that a plane a) Arrives on time given that it has departed on time

b) Departs on time given that it has arrived on time

a) 𝑃 𝐴 𝐷 =𝑃(𝐴∩𝐷)

𝑃(𝐷)=

0.78

0.83= 0.94

b) 𝑃 𝐷 𝐴 =𝑃(𝐷∩𝐴)

𝑃(𝐴)=

0.78

0.82= 0.95

Ms Nurazrin Jupri

• For dependent events A and B, 𝑃 𝐴 ∩ 𝐵 = 𝑃 𝐴 . 𝑃 𝐵 𝐴

• For independent events A and B, if and only if

a) 𝑃 𝐵 𝐴 = 𝑃 𝐵 b) 𝑃 𝐴 𝐵 = 𝑃 𝐴

c) 𝑃 𝐴 ∩ 𝐵 = 𝑃 𝐴 . 𝑃(𝐵)

Probability of Events

Multiplicative Rule

Ms Nurazrin Jupri

Example 13

• A box contain 7 red chips and 3 white ones. 2 chips randomly selected from the box. Find the probability that we get a red chip on the first draw and a white one on the second draw if a) The chips are selected without replacement

b) The chips are selected with replacement

Let R=red chips, W=white chips; a) 𝑃 𝑅 ∩ 𝑊 = 𝑃 𝑅 . 𝑃 𝑊 𝑅

=7

10×

3

9=

7

30

a) 𝑃 𝑅 ∩ 𝑊 = 𝑃 𝑅 . 𝑃 𝑊 𝑅

=7

10×

3

10=

21

100

Ms Nurazrin Jupri

Exercise 3

1. A jar contains black and white marbles. Two marbles are chosen

without replacement. The probability of selecting a black marble

and then a white marble is 0.34, and the probability of selecting a

black marble on the first draw is 0.47. What is the probability of

selecting a white marble on the second draw, given that the first

marble drawn was black?

2. Two cards are drawn one after another from a pack of 52 playing

cards. The first card is replaced before drawing the second card.

What is the probability of getting a ‘king’ in the second card?

Ms Nurazrin Jupri

3. There are 4 red pens and 5 blue pens in a box. Two pens are to

be withdrawn one after another successively. However, the first

pen drawn is replaced. What is the probability of drawing:

a) the first pen is red and the second pen is blue?

b) one red pen and one blue pen?

4. A bag contains 3 red balls and 4 blue balls. One ball is

withdrawn, the color noted and the ball is not replaced. A

second ball is then withdrawn. What is the probability that the

second ball is red?

Exercise 3 (cont.) Ms Nurazrin Jupri

Solving probability using Contingency

Table

Example 14

The management of a computer shop recorded the sales of 250 units of computers in one

month. Of those 250 units sold, 180 units belonged to Pentium II model, of which 30 units

were sold with 17 inch monitors. 25 units of Pentium I model were sold with 17 inch monitors.

The rest of the units were sold with 15 inch monitors. Find the probability that a unit sold,

selected at random,

▫ a) has a 15 inch monitor

▫ b) is a Pentium I model with a 15 inch monitor

▫ c) either is a Pentium II model or has a 15 inch monitor

Ms Nurazrin Jupri

15 inch monitor 17 inch monitor

Pentium I 45 25 70

Pentium II 150 30 180

195 55 250

Ms Nurazrin Jupri