Probability (Elective)

An area of mathematics PROBABILITY THEORY, provides a measure of the likelihood of the outcome of phenomena and events.

Insurance companies use it to decide on financial policies, the government uses it to determine its fiscal and economic policies, theoretical physicists use it to understand the nature of atomic – sized systems in quantum mechanics, and public – opinion polls.

Problem:   A spinner has 4 equal sectors colored

yellow, blue, green and red. What are the chances of landing on blue after spinning the spinner? What are the chances of landing on red?

Solution:   The chances of landing on blue are 1 in 4,

or one fourth.  The chances of landing on red are 1 in 4, or

one fourth.

DEFINITION

Probability is the measure of how likely an event is.

The probability of landing on blue is one fourth.

An experiment is a situation involving chance or probability that leads to results called outcomes.

Example, the experiment is spinning the spinner.

An outcome is the result of a single trial of an experiment.

Example, the possible outcomes are landing on yellow, blue, green or red.

An event is one or more outcomes of an experiment. It is the subset of the sample space

Solution: One event of this experiment is landing on

blue.

Sample point space is the set of all possible outcomes of an experiment. Each element of the sample space is called sample point or simple outcome.

Example: 1. If the experiment is tossing a coin, the sample space is {heads, tails}. 2. If the experiment is drawing a card from a bridge deck, one sample space is the set of cards. 3. If the experiment is tossing a coin twice, a sample space is {HH, HT, TH, TT}

Probability Of An Event

P(A) = The Number Of Ways Event A Can Occur

The total number Of Possible Outcomes

The probability of event A is the number of ways event A can occur divided by the total number of possible outcomes. 

 Experiment 1:   A spinner has 4 equal sectors colored

yellow, blue, green and red. After spinning the spinner, what is the probability of landing on each color?

Outcomes:   The possible outcomes of this experiment

are yellow, blue, green, and red.

Probabilities:  

P(yellow)  = # of ways to land on yellow = 1

total # of colors 4  

P(blue) = # of ways to land on blue = 1 total # of colors 4 

P(green) = # of ways to land on green = 1  total # of colors 4  

P(red) = # of ways to land on red = 1 total # of colors 4 

Experiment 2:   A single 6-sided die is rolled. What is the

probability of each outcome? What is the probability of rolling an even number? of rolling an odd number?

Outcomes:   The possible outcomes of this experiment

are 1, 2, 3, 4, 5 and 6.

P(1) = # of ways to roll a 1= 1 total # of sides 6

P(2) = # of ways to roll a 2 = 1 total # of sides 6

P(3) = # of ways to roll a 3 = 1 total # of sides 6

  P(4) = # of ways to roll a 4 = 1 total # of sides 6

P(5) = # of ways to roll a 5 = 1 total # of sides 6

P(6) = # of ways to roll a 6 = 1 total # of sides 6

P(even) = # ways to roll an even number= 3= 1

total # of sides 6 2

P(odd) = # ways to roll an odd number = 3= 1

total # of sides 6 2

SEATWORK:

I. Give a sample space for each of the following experiments:

1. Selecting a person from an elective class (give two sample spaces)

2. Answering a true – false question 3. Selecting a letter at random from the

English alphabet 4. Tossing a single die 5. Tossing a coin three times 6. Selecting a day of the week

II. Consider the experiment of drawing a numbers 1 through 10,

7. Give the sample space 8. The event of drawing an odd number is the subset ___________ 9. The event of drawing an even number is

the subset _________ 10. The event of drawing a prime number is the subset __________

1. A die is thrown once. What is the probability that the score is a factor of 6?A. 1/6 C. 2/3B. ½ D. 1

2. The diagram shows a spinner made up of a piece of card in the shape of a regular pentagon, with a toothpick pushed through its center. The five triangles are numbered from 1 to 5. The spinner is spun until it lands on one of the five edges of the pentagon. What is the probability that the number it lands on is odd?

A. 1/5 C. 1/2 B. 2/5 D. 3/5

3. Each of the letters of the word MISSISSIPPI are written on separate pieces of paper that are then folded, put in a hat, and mixed thoroughly. 

One piece of paper is chosen (without looking) from the hat. What is the probability it is an I?A. 4/11 C. 1/3B. 2/5 D.1/4

4. There are 10 marbles in a bag: 3 are red, 2 are blue and 5 are green. 

The contents of the bag are shaken before Maxine randomly chooses one marbles from the bag. 

What is the probability that she doesn't pick a red marbles?

A. 3/10 C. 3/7 B. 2/5 D. 7/10

5. What is the probability that the card is either a Queen or a King in a deck of cards?

A. 4/13B. 2/13C. 1/8D. 2/11

1. The factors of six are 1, 2, 3 and 6, so the Number of ways it can happen = 4There are six possible scores when a die is thrown, so the Total number of outcomes = 6

So the probability that the score is a factor of six = 4/6 = 2/3

2. There are three odd numbers (1, 3 and 5), so the Number of ways it can happen = 3There are five numbers altogether, so the Total number of outcomes = 5

∴ The probability the number is odd = 3/5

3. There are 4 I's in the word MISSISSIPPI, so the Number of ways it can happen = 4There are 11 letters altogether in the word MISSISSIPPI, so the Total number of outcomes = 11

So the probability the letter chosen is an I = 4/11

4. There are 7 marbles that are not red: 2 blue and 5 greenThe Number of ways it can happen = 7The Total number of outcomes = 10

5. There are 4 Queens and 4 Kings, so the Number of ways it can happen = 8There are 52 cards altogether, so the Total number of outcomes = 52

Probabilities may be assigned by observing a number of trials and using the frequency of outcomes to estimate probability.

For example, the operator of a concession stand at a park keeps a record of the kinds of drinks children buy. Her records show the following:

Drink Frequency

Cola 150

Lemonade 275

Fruit Juice 75

500

In order to estimate the probability that a child will buy a certain kind of drink, we compute the relative frequency of each drink.

DRINK FREQUENCY RELATIVE FREQUENCY

Cola 150150 = .30

500

Lemonade 275 275 = .55 500

Fruit Juice 75 75 = .15 500

500 1.00

2. A college has an enrolment of 1210 students. The number in each class is as follows.

CLASS NUMBER OF STUDENTS

Freshman 420

Sophomore 315

Junior 260

Senior 215

3. The owner of a hamburger stand found that 800 people bought hamburgers as follows:

KIND OF BURGER FREQUENCY

Mini burger 140

Burger 345

Big Burger 315