Chapter 9 Probability

Aminata Wai, Abigail Simone, Raven Cook, Sashalee Reid, Minasse

Eshete, Maryanna Sapp

IB Math Studies

November 13, 2018

A. Empirical Probability

❖ An Impossible event which has 0%

chance of happening is assigned a

probability of 0

❖ A certain event which has 100%

chance of happening is assigned a

probability of 1

❖ All other events can be assigned a

probability between 0 and 1

We assign to every event a

number which lies between 0

and 1 inclusive. We call this

number a probability

{0, 0.1, 0.2, 0.3, 0.4, 0.5,ect}

Probability

In the field of probability theory we use mathematics to describe the chance or likelihood

of an event happening.

We apply probability theory in our everyday lives such as politics, economics, sports,ect

Experimental Probability

-Observing the results of an experiment

In experiments involving chance we use the following terms to talk about we are doing

and the results we obtain:

❖ The number of trials is the total number of times the experiment is repeated

❖ The outcomes are the different results possible for one trail of the experiment

❖ The frequency of particular outcomes is the number of times that this outcome is

observed

❖ The relative frequency of an outcomes is the frequency of that outcome expressed as

a fraction or percentage of the total number of trials

Exercise 9A.2 pg.264

A. Find the probability that a randomly chosen female 15 year old student at school C is a smoker.

A. Find the probability that a randomly chosen 15 year old student at school E is not a smoker.

A. If a 15 year old is chosen at random from the five schools, what is the probability that he or she is a smoker?

School # of 15 yr olds # of Smokers

Male Female Male Female

A 45 51 10 11

B 36 42 9 6

C 52 49 13 13

D 28 33 9 10

E 40 39 7 4

Total 201 214 48 44

B. Sample Space

SAMPLE SPACE

● A sample space is the set of all possible outcomes of an

experiment.

● It is represented by the universal set U.

● Sample Spaces can be illustrated through Lists, 2- Dimensional

Grids, and Tree Diagrams.

Ex(List). What is the sample space of a rolled die?

U={1,2,3,4,5,6}

2- Dimensional Grids

Ex(2-Dimensional Grid). What is the sample space when two coins

are tossed into the air?

U= {HH, HT, TH, TT}

Tree Diagrams

Ex(Tree Diagram). What is the sample space when a coin is tossed

and a die is rolled?

U={H1, H2, H3,

H4, H5, H6, T1,

T2, T3, T4, T5,

T6}

Practice Problems

Example 1: What is the sample space for twirling a spinner labeled A, B, C, and

D in the form of a list? U={A, B, C, D}

Example 2: Lily has one blue, one black, and one red shirt. She also has one gray

and one blue jeans. How many ways are there for her to dress up in the form of 2-

D grid or tree diagram?

6 Outcomes

C. Theoretical Probability

What is Theoretical Probability?

● What we theoretically expect to occur based on logical reasoning

or the use of a formula.

● Can be written as the ratio of the number of favorable events/

number of possible events (sample space)

P(A)= A desired event / U sample space

Quick Example

If you have two raffle tickets and 100 tickets were sold, what are the

chances of you winning?

● First use the formula P(A)= n(A)/n(U)

● Then, solve for P(A)

Answer: 2/100= .02

A ticket is randomly selected from a basket containing 3 green, 4 yellow, and

five blue tickets. Determine the probability of getting:

a. A green ticket

b. A green or yellow ticket

c. An orange ticket

d. A green, yellow, or blue ticket

1st find the sample space, then calculate the probability of the desired result.

Harder Example

What is the theoretical probability of rolling a 4 or 7 with the set of two

dice?

[1][1], [1][2], [1][3], [1][4], [1][5], [1][6],

[2][1], [2][2], [2][3], [2][4],[2][5], [2][6],

[3][1], [3][2], [3][3], [3][4], [3][5], [3][6],

[4][1], [4][2], [4][3], [4][4], [4][5], [4][6],

[5][1], [5][2], [5][3], [5][4], [5][5], [5][6],

[6][1], [6][2], [6][3], [6][4], [6][5], [6][6].

1st find sample space. 6 possible outcomes

on a single dice, so if there are two you must

multiply to calculate the total number of

possible reasons.

Now, find outcomes that equal to 4 or 7.

The amount of numbers is the number of

favorable outcomes.

P(A)= 9/36= .25 or 25%

Complementary events

Two events are complementary if exactly one of the events must occur.

Example: If A is an event, then A’ is the complementary event of A, or ‘not

A’.

P(A) + P(A’) = 1

Example Problem

5 a. List the six different orders in which Anitta, Kai, and Neda may sit in a

row.

b. If the three of them sit randomly in a row, determine the probability that:

i. Anitta sits in the middle ii. Anitta sits at the left end. iii. Anitta does not sit

at the right end. iv. Kai and Neda are seated together.

Using Grids to find Probabilities

Draw the grid of the sample space when a 5-cent and 10-cent coin are

tossed simultaneously. Hence determine the probability of getting:

a. Two heads b. Two tails c. exactly one head d. At least one head

Table of outcomes

Represents the possible outcomes of a player’s turn using a two-dimensional grid in which the

sum is written at each grid point.

a. Draw a table of outcome to display the possible results when two dice are rolled

and the scores are added together.

b. Hence determine the probability that the sum of the dice is:

i. 11 ii. 8 or 9 iii. Less than 6