PROBABILITYMathematics is always thought of as being a study of certainties – the answer is either right or wrong. But there is a type of mathematics which deals with uncertainties. Since much of our everyday life deals with uncertainties, mathematics has a method for describing these events. This branch of mathematics is called probability theory.

PROBABILITY VOCABULARY Compound events Dependent events Event Experimental probability Independent events Outcome Probability Random Sample space Theoretical probability

Visit the website listed below for definitions of these words:

www.mathwords.com

PROBABILITY SCALE

The probability that an event will happen is somewhere between 0 and 1. When an event has no chance of occurring, we say its probability is zero. When an event is certain to occur, we say its probability is one (or 100%).

Probabilities are usually written as a fraction or whole number is simplest form.

HOW TO FIND SIMPLE PROBABILITY

To find the probability of an event when all outcomes are equally likely, use the formula:

P = number of favorable outcomes

number of possible events

EXAMPLE

A standard deck of cards contains 52 cards. There are four groups of 13 cards. Each group is called a suit. Two of these suits are red and two are black. Each suit contains an ace, king, queen, jack, and the numbers 10 through 2.

If you draw a card at random from the deck of cards, the probability of drawing a heart is…

Because there are 13 hearts & 52 total cards.

4

1

52

13or

COMPOUND EVENTS

A compound event consists of two or more simple events.

To find the probability of compound events, you find the probability of each event and multiply them.

The probability of drawing an Ace from a deck of cards AND rolling an odd number on a standard die is…

26

1

104

4

2

1

52

4

INDEPENDENT EVENTS

When one event does not affect the others, we say that these are independent events.

The probability of getting four tails in a row when a coin is tossed four times is 1 out of 16.

These events are independent since one toss of the coin does not affect the outcome of the next toss.

There are two possible outcomes, heads or tails, so the probability of getting a tail on each toss is ½.

The probability of getting four tails:16

1

2

1

2

1

2

1

2

1)4( tailsP

PRACTICE PROBLEM #1

At Reyna High School 50% of the students eat lunch in the cafeteria.  In the same school 10% of the students participate in sports.  What is the probability that a student selected at random eats in the school cafeteria and participates in sports?

A. C.

B. D.

2

1

10

1

20

1

60

1Answer

DEPENDENT EVENTS

If the outcome of an event does affect the outcome of another event, we say that these are dependent events.

Rick takes two marbles from a bag containing 3 red, 4 blue, 5 green, & 2 yellow marbles.

What is the probability that both marbles are red?

P(2 red) = P(red on 1st draw) × P(red on 2nd draw)

91

3

182

6

13

2

14

3P

PRACTICE PROBLEM #2

A jar contains 6 red marbles and 10 blue marbles, all of equal size. If Dominic were to randomly select 1 marble without replacement and then select another marble from the jar, what would be the probability of selecting 2 red marbles from the jar?

A. C.

B. D.

64

9

8

1

5

3

8

3

Answer

ADDITIONAL RESOURCES

Additional resources for finding probability of compound events:

http://www.mathgoodies.com/lessons/vol6/independent_events.html

THEORETICAL PROBABILITY

Theoretical probability is the likeliness of an event happening based on all the possible outcomes.

The ratio for the probability of an event 'P' occurring is:

P(event) = number of favorable  outcomes number of possible outcomes.

Examples of Theoretical Probability From the letters A, E, I, O, U the theoretical

probability of selecting the letter E is . 5

1

PRACTICE PROBLEM #3

A coin is tossed on a standard 8×8 chessboard. What is the theoretical probability that the coin lands on a black square?

A. 0.5B. 0.25 C. 0.42 D. 0.6

Answer

EXPERIMENTAL PROBABILITY

Experimental probability of an event is the ratio of the number of times the event occurs to the total number of trials.

Examples of Experimental Probability Sam rolled a number cube 50 times. A 3

appeared 10 times. Then the experimental probability of rolling a

3 is 10 out of 50 or 20%.

http://www.northstarmath.com/sitemap/experimentalprobability.html

PRACTICE PROBLEM #4

A coin is tossed 60 times. 27 times head appeared. Find the experimental probability of getting heads.

A. C.

B. D.

27

1

20

9

60

1

20

3

Answer

PRACTICE PROBLEM #5

The table shows the results of a number cube being rolled.

Based on these results, what is the experimental probability of rolling a 1?

A. 2.5% B. 1/6 C. 2/5 D. 0.6

Outcome

Frequency

1 6

2 2

3 2

4 3

5 2

6 0

Answer

PRACTICE PROBLEM #6

The table shows the results of rolling a fair number cube 50 times.

What is the difference between the theoretical probability of rolling a number less than 4 and the experimental results recorded in the table above?

A. 8%C. 58% B. 79% D. 29%

Outcome

Frequency

1 7

2 12

3 10

4 8

5 8

6 4

Answer

USING PROBABILITY TO MAKE PREDICTIONS

Since history tends to repeat itself, probability is used to make predictions.

To make predictions with probability, you set up a proportion.

Example: Last basketball season, John made 60% of the free throws he attempted.

In his 1st game this season, he went to the free-throw line 8 times.

How many free throws did John make if his success rate from last year continued?

See answer on next slide.

ANSWER TO EXAMPLE PROBLEM

John would have made about 5 of his 8 free throws in order for his success rate to continue.

Another way to solve… Convert 60% to a decimal number and multiply times 8:

60% = .6 × 8 = 4.8

8.4

4801008100

60

x

x

x

PRACTICE PROBLEM #7

The student council surveyed a random sample of students and asked which of the following activities the students would prefer as a school trip.

There are 2,340 students in the school.

Based on the data in the survey, how many students are likely to choose skating?A. 234 C. 351B. 260 D. 468

Answer

PRACTICE PROBLEM #8

The probability of a table-tennis ball being

defective is . About how many balls would

be defective in a case of 725 table-tennis

balls?

A. 1

B. 7

C. 73

D. 80

10

1

Answer

ADDITIONAL PRACTICE

Now, log in to Study Island at www.studyisland.com.

Click on “My Classes” in the left-hand side bar. Choose my class, if necessary. Select the assignment titled “Probability” and

work 10 problems in test mode. I will check Study Island to see who

completed the assignment and, YES, it is for a grade!

When finished w/ Study Island, you may play probability games listed on http://classroom.jc-schools.net/basic/math-prob.html.

End Show

ANSWER TO PROBLEM #1

Correct answer: C

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ANSWER TO PROBLEM #2

Correct answer: B

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ANSWER TO PROBLEM #3

Correct answer: A Solution: Step 1: Theoretical probability = number of

favorable outcomes / number of possible outcomes.

Step 2: The probability of the coin lands on the black square is 32.

Step 3: Total number of outcomes = 64. Step 4: P (event) = 5.0

2

1

64

32

Back

ANSWER TO PROBLEM #4

Correct Answer: B Solution: Step 1: Experimental probability = # of times

the event occurs ÷ total # of trials Step 2: # of times a head appears = 27. Step 3: Total # of experiments = 60 Step 4: So, the experimental probability of

getting a head is…                       

20

9

60

27

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ANSWER TO PROBLEM #5

Correct Answer: C

Six out of 15 rolls resulted in a 1, which simplifies to 2 out of 5.

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ANSWER TO PROBLEM #6

Correct answer: A

Number times a number less than 4 was rolled in the experiment = 29

Number times a number less than 4 should theoretically be rolled = 25 (1/2 of 50)

Difference: 29 – 25 = 4 4 ÷ 50 = .08 = 8%

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ANSWER TO PROBLEM #7

Correct answer: A

200 of the 2,340 students were surveyed.

Of those 200, 20 would prefer to go skating. There are a couple of ways to solve this…

1) 2) (20 ÷ 200) × 2340 = 234

234

468002002340200

20

x

x

x

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ANSWER TO PROBLEM #8

Correct answer: H

(1/10) × 725 = 72.5

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