Grade 11 Day 16 Probability worksheets

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Matrics:

Please do revision of Gr11 Probability. It will be part of the June exam (± 10

marks in paper 1)

Work through the examples and please do the exercises from p7.

Please use this time to study your gr11 work, we don’t know how much time

you are going to have to study for the exam.

Remember that the exam work and previous papers are available on Moodle.

http://e-learn.gc.co.za/moodle/course/view.php?id=214

http://e-learn.gc.co.za/moodle/course/view.php?id=214

Grade 11 Probability

Grey College 2

Gr10 Revision

The probability of an event occurring always lies between 0 and 1.

0 is impossible event

1 is a certain event

The probability of an event occurring always lies between 0 and 1.

0 is impossible event

1 is a certain event

1.1.1 Notation

P(A) – The probability of event A occurring

P( A) = P(not A) – The probability of an event A not occurring

or complement of A

P(A or B) P(A B) – The probability of event A or B occurring -

or means union

P(A and B) P(A B) – The probability of event A and B occurring

and means intersection

Conditional probability.

In this situation, the probability of event A is dependent on the outcome of the probability of event B.

In other words, event B has already happened, now what is the probability of event A?

1.1.2 Mutually exclusive events

Two events are mutually exclusive if the two events cannot occur in the same trial.

e.g. Consider a trial is the tossing of a dice, the event A is obtaining a 6 while the event B is obtaining a 2, it

is impossible to have both events A and B together.

For mutually exclusive events : P(A and B) = 0

Addition rule states that P(A or B) = P(A) + P(B) – P(A and B)

But for mutually exclusive events: P(A or B) = P(A) + P(B) because P(A and B) = 0

Complementary events

Two events are complementary if P( A ) = 1 – P(A) or P(not A) = 1 – P(A)

There are two conditions that have to be true for events A and B to be complementary:

Grade 11 Probability

Grey College 3

1. The events have to be mutually exclusive

2. P(A) + P(B) =1

Two events A and B are complementary if P( A ) = 1 – P(A) = P(B)

Example

1. Determine whether or not the following events are complementary given a sample space

S = {6; 7; 8; 9; 10}.

1.1 Selecting an even number and a multiple of 5.

1.2 Selecting a multiple of 3 and a prime number.

1.3 Selecting an odd number and a multiple of 2.

Solution

1.1 Let the set of even numbers be given by A={6; 8; 10} and

the set of multiples of 5 be represented by B={10}.

Remember the two conditions

These two events share a common element, 10

They are not complementary events

1.2 Let the set of multiples of 3 be D ={6; 9} and set of prime numbers be E= {7}.

No common element

Then what is P(D) + P(E)?

P(D) + P(E)= 51

52

=53

P(D) + P(E) 1

The events are not complementary

1.1.3 Dependent and independent events

Dependent events- NO REPLACEMENT

Events L and event G are dependent if the outcome of event L has an influence on the outcome of event G.

The occurrence of event L changes the probability of event G.

e.g. We draw a card from a pack of cards and draw a second card without replacing the first card.

Grade 11 Probability

Grey College 4

Independent events – replacement

Event L and event G are independent if the outcome of event L has no influence on the outcome of event G.

The occurrence of event L has no effect on the probability of event G.

e.g. We draw a card from a pack of cards and draw a second card after replacing the first card.

Product rule

For independent events : P(A and B) = P(A)P(B)

Example

1. Given the following probabilities: P(A) = 0,45 , P(B) = 0,3 and P(A or B) = 0,615.

1.1. Determine whether or not events A and B are mutually exclusive.

1.2. Determine P(A and B).

1.3 Are events A and B independent?

Solution

1.1 P(A or B) = P(A) + P(B) – P(A and B)

0,615 = 0,45 + 0,3 – P(A and B)

P(A and B)= 0,75 – 0,615

= 0,135

The events are not mutually exclusive.

1.2 Answered above

P( A or B) = P(A) + P(B) – P(A and B)

0,615 = 0,45 + 0,3 – P(A and B)

P(A and B) = 0,75 – 0,615

= 0,135

1.2 P(A and B) = P(A)P(B) if the events are independent.

P(A)P(B) = 0,45 0,3

= 0,135

and

P(A and B) = 0,75 – 0,615 was calculated above

= 0,135

The events are independent because P(A and B) = P(A)P(B).

Grade 11 Probability

Grey College 5

1.1.4 Contingency tables

Contingency tables are statistical tables that show the relationships between two or more variables. They are

often used to determine whether or not events are independent.

Examples

1. Researchers at Stella Hospital found that many people are carriers of a certain disease.

They tested 200 people and found that 5 of the 90 females tested were carriers, and that 7 of

the males tested were carriers. Determine:

1.1 the probability that a male from those tested is a carrier

1.2 the number of males in South Africa who are likely to be a carrier (assuming that the population of

South Africa is 50 million)

1.3 the probability that the person tested is carrier, given that the person is a female

1.4 the probability that the person tested is male, given that is a carrier.

Solution

Two factors, male/females and carriers/non-carriers are complementary even.

Carrier Non-

Carrier

Males 7 103 110

Females 5 85 90

12 188 200

1.1 2007

)( CMP

1.2 000750100050002007

1.3 P(C|F)905

1.4 P(M|C))(

)(

CP

CMP

127

1.3 Let the set odd numbers be L={7; 9} and the set of multiples of 2 be G ={6; 8; 10}.

No common element

P(L) + P(G) = 53

52

= 55

= 1

The events are complementary.

Grade 11 Probability

Grey College 6

2. Determine whether or not the following events are complementary given a sample space

S = {6; 7; 8; 9; 10}.

2.1 Selecting an even number and a multiple of 5.

2.2 Selecting a multiple of 3 and a prime number.

2.3 Selecting an odd number and a multiple of 2.

Solution

2.1 Let the set of even numbers be given by A={6; 8; 10} and

the set of multiples of 5 be represented by B={10}.

Remember the two conditions

These two events share a common element, 10

They are not complementary events

2.2 Let the set of multiples of 3 be D ={6; 9} and set of prime numbers be E= {7}.

No common element

Then what is P(D) + P(E)?

P(D) + P(E)= 51

52

=53

P(D) + P(E) 1

The events are not complementary.

2.3 Let the set odd numbers be L={7; 9} and the set of multiples of 2 be G ={6; 8; 10}.

No common element

P(L) + P(G) = 53

52

= 55

= 1

The events are complementary

3.

If A and B are

independent events,

find the values of x

and y.

0,3 0,2

A

x

y

Grade 11 Probability

Grey College 7

Worksheet 1

QUESTION 4

4.1 A researcher randomly selects 1000 death certificates and, after interviewing the attending

physician, records the following information about the deceased.

Cancer Heart Disease Other

Smoker 135 310 205 650

Non-smoker 55 155 140 350

190 465 345 1000

4.1.1 What is the probability that all deaths attributed to cancer?

(2)

4.1.2 Is Smoking and Cancer as Cause of Death, dependant or not? Explain your answer.

(4)

4.2 A bag contains five red and seven green marbles. Two marbles are drawn successively

without replacement. Calculate the probability that both marbles are the same colour.

(6)

4.3 In a study 170 Grade 11 learners were asked if they take Biology, Accounting or Science.

The result of the study are shown below:

70 people take Biology

80 people take Accounting

80 people take Science

43 people take Biology and Science

36 people take Accounting and Science

136 people take at least one of the subjects

15 take all 3 subjects

4.3.1 Record the above information in a Venn-diagram (5)

4.3.2 How many people take none of the subjects? (1)

4.3.3 How many people take Biology and Science but NOT Accounting? (1)

4.3.4 What is the probability that a person chosen at random takes at least two of the

subjects?

(2)

[21]

Grade 11 Probability

Grey College 8

Worksheet 2

Grade 11 Probability

Grey College 9

Worksheet 3