IB Math SL Year 2

Name: ________________________Date: _________________________

Lesson 2-6: Probability, Venn Diagram, Combined and Disjoint Events

In this lesson we will revisit the following learning goals:

1. What is probability? What are the basic rules of probability? 2. How do we create a Venn Diagram? What steps should we follow?

3. How can we determine probabilities involving two events, A and B, using P (A∪B )=P ( A )+P (B )−P (A ∩B)?

What does it mean for events to be mutually exclusive?

Probability formulas can be found on page _______ in our formula booklets!

This means: Probability=number of outcomes youwant

totalnumber of outcomes

Complements--The set of outcomes that are NOT in the event A is called the complement of A, denoted A’

This means that when you are looking for A’ (probability that something will not happen) we subtract)

For instance:o The probability that it will rain is 0.6. The probability that it will not rain is 1 - 0.6 = 0.4.

When two events are being discussed, always use this formula (unless the problem states mutually exclusive for independent

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Station 1: Probability Probability --- is defined as: the likelihood/chance of something happening

Probability Rules

1. Every probability must be between 0 and 1, inclusive.2. The sum of the probabilities of all possible outcomes of a trial must be 1.

Probability Number Line

List the members in your group for each role:

____________________ Instructional Coach

____________________ Progress Coach

____________________ Collaboration Coach

0 0.5 1

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Some Examples:

1. The table below shows the relative frequences of the ages of the students at a high school.a. A student is randomly selected from the school. Find the,

i. The student is 15 years old.

ii. The student is 16 years of age or older.

There are 1200 students at this high school.b. Calculate the number of 15 year old students.

Try one together:

2. The number of club members of each gender choosing each game in a particular year is shown in the table below.

Billiards Snooker DartsMale 39 16 8Female 21 14 17

A club member is to be selected at random. What is the probability that the club member selecteda. Is female and likes darts? b. Is a male?

Station 2: Venn Diagrams

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A

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A sample space is a list of all possible outcomes. The set of ALL possible outcomes is what we call the Universal Set,notation ______

Example: Let U={1 ,2 ,3 ,4 ,5 ,6 ,7 ,8}.Let A={1 ,3 ,4 ,5 ,6 } andB={evennumbers∈U }Represent this information in a Venn diagram.

We often use Venn Diagrams to represent probabilities or the number of elements in sets.

Steps to CREATE a VENN DIAGRAM1. Draw a BOX to represent the UNIVERSAL SET2. 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!

READING a Venn DiagramYou can identify the union, intersection, and complement of events in a Venn Diagram.

You try:

B

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3. Suppose there is a class of 100 senior students.38 of the seniors are IB Diploma Candidates.Of the 100 seniors, 30 are student athletes.16 are both diploma candidates and student athletes.Show this in a Venn diagram.

4. From our Venn Diagram above… a) The probability that a student chosen at random is both a diploma candidate and a student athlete is

written P(A∩B), what is this probability?

b) Find the probability that a randomly selected senior is not a student athlete but is a diploma candidate.

c) Find the probability that a randomly selected senior is not a student athlete and is not a diploma candidate.

d) Find the probability that a randomly selected senior is only an athlete.

e) The probability that a student chosen at random is a diploma candidate or a student athlete is written P(A∪B), what is this probability?

f) Find and interpret P(A∪B').

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Station 3: Combined and Disjoint EventsFormula Page:

Use this rule when: we are given 3 out of the 4 components.

Use this rule when: told or asked for “mutually exlusive.”

*Combined events:

5. In a group of 80 tourists 40 have a camera, 50 are females, and 22 are females with cameras. What’s the probability someone is a female or has a camera?

P(cam.) = 4080 P(C∪F) = P (C )+P (F )−P (C ∩F )

P(F) = 5080 P(C∪F) = ( 4080 )+( 5080 )−( 2280 ) =

6880 =

1720

P(Cam.∩ F )= 2280

*Mutually Exclusive:

Two events are said to be Mutually Exclusive if - they cannot occur at the simultaneously ex. sitting in math class and English at 10am.

- they cannot both happen

Find the following for mutually exclusive C and D:

- n(C ∩D) = 0 P (C∩D )=0P (C∪D )=P ( A )+P (B)

They cannot happen together!Let’s Try: 6. Of the 1260 households in a small town, 632 have dogs, 568 have cats, and 114 have both types of pets.

a) Construct a Venn diagram to display this.

b) If a household is chosen at random, what is the probability that the household has neither a cat nor a dog?

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c) Are the events having a cat and having a dog mutually exclusive? Justify with mathematics. Station 4: Mixed Practice

d) A used car dealer has 150 used cars on his lot. The dealer knows that 30 of the cars are defective. One of the 150 cars is selected at random.

a. What is the probability that it is defective?

b. What is the probability that is it not defective? Show your work using the formula for complimentary events.

e) A survey of 400 people is carried out by a market research organization in two different cities, Buenos Aires and Montevideo. The people are asked which brand of cereal they prefer out of Chocos, Zucos or Fruti. The table below summarizes their responses.

Chocos Zucos Fruti TotalBuenos Aires 43 85 62 190Montevideo 57 35 118 210Total 100 120 180 400

(a) One person is chosen at random from those surveyed. Find the probability that this person does not prefer Zucos.

(b) What’s the probability someone lives in Buenos Aires and likes Fruti cereal?

f) A box contains 10 coloured light bulbs, 5 green, 3 red and 2 yellow. One light bulb is selected at random and put into the light fitting of room A. What is the probability that the light bulb selected is

- green?

- not green?

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g) A basket of fruit contains 10 apples, 6 bananas and 4 oranges. A fruit is selected at random. a) Find the probability that the first fruit is not orange.

b) Find the probability you pick an apple or an orange.

h) The following chart was used to show what type of movie King High School students prefer best.

Action Romance Historical TotalMale 10 17 8 35Female 3 4 8 15Total 13 21 16 50

a) If a student is selected at random, what’s the probability the student is a male and likes action best? (Remember, AND means the intersection/overlap!)

b) Of the males , what’s the probability that they like action movies?

i) In a group of 30 students, 17 play computer games, 10 play board games, and 9 play neither. Find the probability that a randomly selected student:

a. plays board games,

b. plays both computer games and board games,

c. plays board games but not computer games

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j) Let U = {2, 3, 4, 5, 7, 9}X = {x : x ∈Z, 2≤x<6}Y = {x : x ∈ odd numbers}Z = {2, 4, 5, 7, 9}

a) What elements belong to X ∩Y ?

b) Is 5∈(X ∩Y )?

c) What is n(X∪Z)?

d) Is Y ⊆Z? Explain why or why not.

k) In a group of 35 children, 10 have blonde hair, 14 have brown eyes, and 4 have both blonde hair and brown eyes.

a. Draw a Venn Diagram to represent this situation.

b. A child is selected at random. Find the probability that the child has blonde hair or brown eyes.

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l)

m) If A and B are two events such that P (A )= 920 and P (B )= 3

10 and P (A∪B )=2P(A∩B), find

a. P (A∪B )

b. P (A∪B )'

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c. P(A∩B')

d. Are A and B mutually exclusive? Use math to show why or why not?

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