Transcript 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
An experiment or event is the act of doing something to create a
result.
The possible outcomes of an experiment are all of the different
results that can occur, although usually only one outcome occurs for
each experiment.
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
P(event) 
The total number of possible outcomes
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 Probability
Example:
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?
Procedure:
1. Identify the experiment or event.
2. Identify the total number of possible outcomes.
1
2
4
3
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
P(land on 2) 
4
Bar Graphs and Probability
Example:
The following bar graph represents the 50 final grades in Mr.
Miller’s Statistics class last semester.
Number of Grades
If one student is randomly
selected, find the probability
that the student received a final
grade of an “A” on the final.
Number of Grades
25
Procedure:
Use the bar graph to find the
probability a student received an “A”.
6
3
P(A) 

50 25
20
15
10
5
0
6
A
10
23
8
3
B
C
D
F
Grades
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.
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 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.
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 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.
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 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
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.
Suppose a card is drawn at random
from a regular deck of 52 cards.
What is the probability that the card
is a face card?
Suppose a card is drawn at random
from a regular deck of 52 cards.
What is the probability that the card
is a “one-eyed jack”?
Suppose a card is drawn at random
from a regular deck of 52 cards.
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?