6.4 counting techniques and simple probabilities

6.4 Counting Techniques and Simple Probabilities

A set is a well-defined group of objects or elements. Counting, in this section, means determining all the possible ways the elements of a set can be arranged. One way to

do this is to list all the possible arrangements and then count how

many we have.

Example: List and count the ways the elements in the set A,B,C can

be arranged.

• ABC, ACB

• BAC, BCA

• CAB, CBA

• There are 6 ways A,B, and C can be arranged

If there are more than 3 elements in the set, the procedure by listing

becomes more challenging.

Count the ways W, X, Y, Z can be arranged.

• WXYZ, WXZY, WYZX, WYXZ, WZXY, WZYX

• XWYZ, XWZY, XYWZ, XYZW, XZWY, XZYW

• YWXZ, YWZX, YXWZ, YXZW,YZXW, YZWX

• ZWXY, ZWYX, ZXWY, ZXYW, ZYWX, ZYXW

• 24 ways

To determine the number of ways for arranging a specific number of

items without repetition:• Determine the number of slots to be filled.

• Determine the number of choices for each slot.

• Multiply the numbers from step 2.

Count the ways W, X, Y, Z can be arranged.

• Step 1: There are 4 slots.

• Step 2: 4 choices for slot 1, 3 choices for slot 2, 2 choices for slot 3 and 1 choice for slot 1.

• Step 3: 4x3x2x1 = 24

_____ _____, _____, _____,4 3 2 1

How many ways are possible for arranging containers of cotton balls,

gauze pads, swabs, tongue depressors, and adhesive tape in a row on a shelf in a doctor’s office?

_____ _____, _____, _____, _____,5 4 3 2 1

5 x 4 x 3 x 2 x 1 = 120 ways

Probability is a number that describes the chance of an event occurring if an activity is repeated

over and over. A probability of zero means the event cannot occur

while a probability of one means the event must occur. Otherwise

the probability can be expressed as a fraction, decimal or percent.

The Vocabulary of Probability

The possible outcomes of an experiment are all of the different results that can occur, although usually only one outcome occurs for each experiment.

An experiment or event is the act of doing something to create a result.

A success is the outcome we’re most interested in occurring.

The probability of an event is a ratio, abbreviated as P(event), and is calculated

The number of successes in the event(event)The total number of possible outcomes

P

A random selection is the act of choosing something so that each possible outcome has an equal chance of being selected and is equally likely to be selected.

What is an event?An event is an experiment or collection of

experiments. Examples: The following are examples of

events. 1) A coin toss.(2) Rolling a die.

(3) Rolling 5 dice.4) Drawing a card from a deck of cards.

5) Drawing 3 cards from a deck.6) Drawing a marble from a bag of different

colored marbles.7) Spinning a spinner in a board game.

The following are possible outcomes of events.

• 1) A coin toss has two possible outcomes. The outcomes (sample space) are "heads" and "tails".

• 2) Rolling a regular six-sided die has six possible outcomes. You may get a side with 1, 2, 3, 4, 5, or 6 dots.

• 3) Drawing a card from a regular deck of 52 playing cards has 52 possible outcomes. Each of the 52 playing cards is different, so there are 52 possible outcomes for drawing a card.

The Vocabulary of ProbabilityExample:The following spinner is divided equally into 4 pieces. There are 4 possible outcomes – the spinner can land on 1, 2, 3, or 4. What is the probability the spinner will land on 2?

1 2

4 3

Procedure:1. Identify the experiment or event.2. Identify the total number of possible outcomes.

3. Identify the number of successes described in the event.

The experiment is spinning the spinner; there is a total of 4 possible outcomes; this event has only one success (landing on a 2). 1(land on 2)

4P

Bar Graphs and ProbabilityExample:

The following bar graph represents the 50 final grades in Mr. Miller’s Statistics class last semester.

Nu

mb

er o

f G

rad

esA B DC F

0

5

10

15

20

610 23 8 3

Grades

Number of Grades25

If one student is randomly selected, find the probability that the student received a final grade of an “A” on the final.

Procedure:Use the bar graph to find the probability a student received an “A”.

6 3(A)50 25

P

Suppose there are 10 balls in a bucket numbered as follows: 1, 1, 2, 3, 4, 4, 4, 5, 6, and 6. A single ball is randomly chosen from

the bucket. What is the probability of drawing a ball numbered 1?

There are 2 ways to draw a 1, since there are two balls numbered 1. The total possible number of outcomes is 10, since there are

10 balls. The probability of drawing a 1 is the ratio

2/10 = 1/5.


the bucket. What is the probability of drawing a ball with a number greater than 4?

There are 3 ways this may happen, since 3 of the balls are numbered greater than 4. The

total possible number of outcomes is 10, since there are 10 balls. The probability of drawing a number greater than 4 is the ratio 3/10. Since this ratio is larger than the one in the previous example, we say that this event has a greater

chance of occurring than drawing a 1.


the bucket. What is the probability of drawing a ball with a number greater than 6?

Since none of the balls are numbered greater than 6, this can occur in 0 ways. The

total possible number of outcomes is 10, since there are 10 balls. The probability of

drawing a number greater than 6 is the ratio

0/10 = 0.


the bucket. What is the probability of drawing a ball with a number less than 7?

Since all of the balls are numbered less than 7, this can occur in 10 ways. The total possible number of outcomes is 10, since there are 10 balls. The probability of drawing a number less

than 7 is the ratio 10/10 = 1.Note in the last two examples that a probability of 0 meant that the event would not occur, and

a probability of 1 meant the event definitely

would occur.

Suppose a card is drawn at random from a regular deck of 52 cards.

What is the probability that the card is an ace?

There are 4 different ways that the card can be an ace, since 4 of the 52

cards are aces. There are 52 different total outcomes, one for

each card in the deck. The probability of drawing an ace is the

ratio 4/52 = 1/13.


What is the probability that the card is a face card?


What is the probability that the card is a “one-eyed jack”?


What is the probability that the card is red?

Suppose a regular die is rolled. What is the probability of getting a

3 or a 6? There are a total of 6 possible

outcomes. Rolling a 3 or a 6 are two of them, so the probability is

the ratio of 2/6 = 1/3.

A class has 13 male and 15 female students. If a student is randomly

selected, what is the probability the student is a male?

6.4 counting techniques and simple probabilities

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