Name_________________________________

Date: ____________

Lesson 11-7 Sets, Probability and Basic Statistics

Learning Goals: What must I know, understand and be able to do with sets, probability

and basic statistics?

Sets, Probability and Basic Statistics

Recall the steps to CREATE a VENN DIAGRAM

1. Draw a BOX to represent the UNIVERSAL SET

2. Draw a circle for each set you are given in a problem, do NOT draw a circle for the universal set!

3. Begin by looking at the INTERSECTION of the sets and work your way OUT!

*DON’T SUBTRACT If you see the phrase “but not _____” or “only”

Symbol Name Meaning/Definition

Real numbers All numbers that can be represented by a decimal

ℚ Rational numbers Any number that can be expressed in the form a/b

ℤ Integers Any whole number, positive or negative (includes 0)

ℕ Natural numbers Numbers you use for counting (includes 0)

or Universal set Set of all possible elements

If U = {1,2,3,4,5,6,7,8}, then sets A and B must be in U

Element of Set membership

If A={3,9,14}, then 3 A

Not an element

of No set membership

If A={3,9,14}, then 1 A

Cardinality Number of elements in a set If A = {3, 9, 14}, then n(A) = 3

Complement of a set

All objects that do not belong to a set If U = {1, 2, 3, 4, 5, 6} and A = {1,2}; then A’ = {3, 4, 5, 6}

Subset Every element in the first set is also in the second set *the two sets can be equal*

If A = {2, 3, 5} and B = {1, 2, 3, 4, 5}then A B; but B is not A

If A = {2, 3, 5} and B = {2, 3, 5} then A B.

Ø or { } Empty set (null set)

No elements found in a set *the empty set is subset of all sets*

If A = {3, 9, 14} and B = {1, 2, 4}, then A B = { }

A ∩ B intersection Elements that belong to set A and set B (overlap) If A = {3,7,9,14} and B = {9,14,28}, then A ∩ B = {9,14}

A ∪ B union Elements that belong to set A or set B (everything) If A = {3,7,9,14} and B = {9,14,28},

then A ∪ B = {3,7,9,14,28}

Probability

Basic Statistics

The heights of 200 students are recorded in the following table.

Height (h) in cm Frequency

140 ≤ h < 150 2

150 ≤ h < 160 28

160 ≤ h < 170 63

170 ≤ h < 180 74

180 ≤ h < 190 20

190 ≤ h < 200 11

200 ≤ h < 210 2

(a) Write down the modal group.(1 mark)

(b) Add a column to the table to show the cumulative frequencies. (2 marks) (c) Calculate

i. an estimate of the mean of the heights (1 mark)

ii. an estimate of the standard deviation of the heights, write your answer exactly. (1 mark)

Notes

Notes

There are 120 teachers in a school. Their ages are represented by the cumulative frequency graph below.

a) Determine the median (50th percentile).

b) What is the 40th percentile?

c) What is the 95th percentile?

d) What is the 70th percentile?

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75

Age

130

120

110

100

90

80

70

60

50

40

30

20

10

0

Cum

ula

tiv

e fr

equen

cy

Notes

You Try it! Sets, Probability and Basic Statistics

1) Today Philip intends to go walking. The probability of good weather (G) is . If the weather is good, the

probability he will go walking (W) is . If the weather forecast is not good (NG) the probability he will go

walking is .

a) Complete the probability tree diagram to illustrate this information. (4 marks)

b) What is the probability that the weather is not good and Philip goes walking? (1 mark)

c) What is the probability that Philip will go walking? (3 marks)

d) What is the probability the weather is not good given that Philip went walking? (2 marks)

4

3

20

17

5

1

34

G

NG

NW

NW

W

W

1720

2) The weights in kilograms of 12 students in a class are as follows.

63 76 99 65 63 51 52 95 63 71 65 83

(a) State the mode. [1 mark]

(b) Calculate i. the mean weight; [1 mark]

ii. the standard deviation of the weights, leave your answer exact. [1mark]

3)

(a) In the Venn diagram below, the number of elements in each region is given.

Find n ((P Q) R).

(2)

(b) U is the set of positive integers, +.

E is the set of even numbers.

M is the set of multiples of 3.

(i) List the first six elements of the set M.

(ii) List the first six elements of the set E′ M. (2)

P 2 3 1 Q

4 6

5

9

R

U

4)

5)

6) Events A and B have probabilities P(A) = 0.4, P (B) = 0.65, and P(A ∪ B) = 0.85. a) Calculate P(A B). (2 marks)

b) State with a reason whether events A and B are independent. (2 marks)

c) State with a reason whether events A and B are mutually exclusive. (2 marks)

210 students participated