Transcript Addition and Multiplication Principles of Counting

Standard:
MM1D1a - a. Apply the addition and multiplication principles of counting.
Probability
Addition and
Multiplication
Principles of Counting
What is Probability?
The likelihood that a given event will
occur.
What real life situations would
involve probability?
Empirical Probability
A scientific guess or estimate
that an event will happen, based
on a frequency of a large
number of trials.
Ex. A cone shaped paper cup is tossed 1,000
times and it lands on its side 753 times, the
relative frequency which the cup lands on its
side is
We estimate the probability of the cup landing
on it’s side is about
This is a biased object because landing on its
side has a better chance than landing on its
base.
Can you think of an unbiased object?
Theoretical Probability
The number of ways that an event
can occur, divided by the total
number of outcomes in the sample
space.
Prob. of Event
number of ways event can occur
= number of possible outcome s in sample space
1. An outcome is a result of a some activity.
Ex. Rolling a die has six outcomes: 1, 2, 3, 4, 5, 6
2. A sample space is a set of all possible
outcomes for an activity.
Ex. Rolling a die sample space: S = {1,2,3,4,5,6}
Tossing a coin: S = {H,T}
3. An event is any subset of a sample space.
Ex. Rolling a die:
the event of obtaining a 3 is E = {3},
the event of obtaining an odd # is E = {1,3,5}
Example 1
Find the probability of rolling an even # on one
toss of a die. (This is an unbiased object)
E = event of rolling even # = {2,4,6}
Number of ways it can happen: n(E) = 3
S = sample space of all outcomes = {1,2,3,4,5,6}
Number of possible outcomes: n(S) = 6
SO…
Probability of rolling an even number is
Example 2
At a sporting goods store, skateboards are
available in 8 different deck designs. Each deck
design is available with 4 different wheel
assemblies. How many skateboard choices does
the store offer?
Lets make a tree diagram!
Example 2 cont…
What else could you do to find the solution?
Multiply # of deck designs by the number
of wheel assemblies
4•8 = 32 skateboard choices
The Multiplication Counting
Principle
If one event can occur in m ways and
another event can occur in n ways,
then the number of ways that both
events can occur together is m•n.
This principle can be extended to
three or more events.
Addition Counting Principle
If the possibilities being counted can be
divided into groups with no possibilities
in common, then the total number of
possibilities (outcomes) is the sum of
the numbers of possibilities in each
group.
HUH???
Think about it!
Suppose that we want to buy a computer
from one of two makes, Dell and Apple.
Suppose also that those makes have 12
and 18 different models, respectively.
Then how many models are there
altogether to choose from ?
Solution
Since we can choose one of 12 models of Dells
or one of 18 of Apples, there are all
together 12 + 18 = 30 models to choose from.
This is the Addition Principle of Counting.
Choosing one from given models of either
make is called an event and the choices for
either event are called the outcomes of the
event. Thus the event "selecting one from
make Dell", for example, has 12 outcomes.
What’s the Probability?
How would we find the probability of
choosing a Dell?
12 Dells
30 total choices
=
2
5
The probability of choosing a Dell would be:
2
= .4 = 40%
5
Example 3
Codes- Every purchase made on a
company’s website is given a
randomly generated confirmation
code. The code consists of 3
symbols (letters and digits). How
many codes can be generated if at
least one letter is used in each?
Example 3 - Solution
To find the number of codes, find the
sum of the numbers of possibilities for
1-letter codes, 2-letter codes and 3letter codes.
1-letter: 26 choices for each letter and
10 choices for each digit. So 26•10•10 =
2600 letter-digit-digit possibilities.
The letter can be in any of the three
positions , so there are 3•2600 = 7800
total possibilities.
Example 3 - Solution
2-letter: There are 26•26•10 = 6760 letterletter-digit possibilities. The letter can be in
any of the three positions , so there are
3•6760 = 20,280 total possibilities.
3-letter: There are 26•26•26 = 17,576
possibilities
Total Possibilities: 7800 + 20,280 + 17,576 =
45, 656 possible codes