Download Interactive Note Book (INB) Please copy all the notes in INB and I

Probability Notes(INB)
March 24, 2016
Interactive Note Book (INB)
Please copy all the notes in INB
and I will explain and do some
guided and independent practice
after the break and start the review
after this last unit
Cover page
Unit #18
Probability
SOL 8.12
Nallari Math 8 Class
Probability Notes(INB)
March 24, 2016
I Can
Determine the probability of no more than three independent events.
Determine the probability of no more than two dependent events without
replacement.
Compare the outcomes of events with and without replacement.
Nallari Math 8 Class
Probability Notes(INB)
March 24, 2016
Alpha Block Sort
ABC
acute
box plots
DEF
experimental probability
events
frequency
GHI
independent events
histogram
correlation
dependent events
geometrical figures
compound events.
fraction
height
compare
decimal
counting principle
exponents
chance
increase
descending order
ascending order
JKL
MN
likely
meter
kilogram
mean
median
mode
RST
sample space
UVW
OPQ
outcome
probability
occurrence
quadrants
XYZ
y-intercept (b)
theoretical probability
with out replacement
y-axis
replacement
vertices
x-axis
rise(y-axis)
run(x-axis)
slope
translations
reflections
Nallari Math 8 Class
Probability Notes(INB)
March 24, 2016
Unit Summary#18
Nallari Math 8 Class
Probability Notes(INB)
March 24, 2016
Vocabulary
Probability:The chance an event will happen written as number from 0 to 1.It can
expressed in fraction, decimal or percentage Ex : 1/2=0.5=50%
P(event) = number of ways event can occur
total number of outcomes
Out come: In probability, the outcome is the result of performing an experiment. The
probability of an event occurring is the ratio of the desired outcomes to the total number
of possible outcomes.
o The probability that an event is likely to occur is close to one.
o The probability that an event is not likely to occur is close to zero.
o The probability that an event is as likely to occur as it is not to occur is close to one half
Experiment
Outcomes
1.Tossing a coin
heads, tails
2.Rolling a six-sided die
1, 2, 3, 4, 5, 6
3.Tossing a coin AND rolling a six-sided die
Outcomes -(heads, 1), (heads, 2), (heads, 3), (heads, 4), (heads, 5), (heads, 6), (tails, 1), (tails, 2), (tails, 3), (tails, 4), (tails, 5), (tails, 6)
Compound events:Events that contain more than one outcome are called compound
events.Two types of compound events 1.Independent and 2.Dependent events.
Independent events: If the outcome of one event does not influence the occurrence of
the other event, they are called independent. If events are independent, then the second
event occurs regardless of whether or not the first occurs. For example, the first roll of a
number cube does not influence the second roll of the number cube. Other examples of
independent events are, but not limited to: flipping two coins; spinning a spinner and
rolling a number cube; flipping a coin and selecting a card; and choosing a card from a
deck, replacing the card and selecting again.
The probability of three independent events is found by using the following
formula: P(A and BandC) = P(A).P(B).P(C)
Ex: When rolling three number cubes simultaneously, what is the probability of rolling a 3 on one cube, a 4 on one cube,
and a 5 on the third?
P(3and4and5)=P(3).P(4).P(5)=1/6 .1/6.1/6 =1 /216
Dependent events:If the outcome of one event has an impact on the outcome of the
other event, the events are called dependent. If events are dependent then the second
event is considered only if the first event has already occurred. For example, if you are
dealt a King from a deck of cards and you do not place the King back into the deck
before selecting a second card, the chance of selecting a King the second time is
diminished because there are now only three Kings remaining in the deck. Other
examples of dependent events are, but not limited to: choosing two marbles from a bag
but not replacing the first after selecting it; and picking a sock out of a drawer and then
picking a second sock without replacing the first.
The probability of two dependent events is found by using the following formula
P(A and B) = P(A).P(B after A)
Ex: You have a bag holding a blue ball, a red ball, and a yellow ball. What is the probability of picking a blue ball out of the
bag on the first pick then without replacing the blue ball in the bag, picking a red ball on the second pick?
P(blue and red) = P(blue) . P(red after blue) = 1/3 . 1/2 =1/6
Nallari Math 8 Class
Probability Notes(INB)
March 24, 2016
Vocabulary
• Counting Principle – a method used to compute the number of possible outcomes of an
experiment. If each outcome has independent parts, the total number of possible outcomes
can be found by multiplying the number of choices for each part.
• Experimental probability – The estimated probability of an event; obtained by
dividing the number of successful trials by the total number of trials.
• Replacement – replacing/ returning an item back into the sample space after an
event and thus allowing an item to be chosen more than once.
• Sample space – the set of all possible outcomes of a probability experiment.
Two six-sided dice are rolled.
If (3, 5) means the first die shows 3 and the second die shows 5, then sample space is
• Theoretical probability – the theoretical probability of an event, P (event), is the
ratio of the number of outcomes in the event to the number of outcomes in the
sample space, if all outcomes are equally likely.
•
P (event)= Number of possible favorable outcome
Total number of possible outcomes
Nallari Math 8 Class
Probability Notes(INB)
March 24, 2016
Videos to watch
https://www.teachingchannel.org/videos/teaching-dependent-and-independent-events
http://www.virtualnerd.com/middle-math/probability-statistics/probability
Videos in Probability and Statistics
Probability
Finding Outcomes
Probability of Independent Events and Areas
Samples and Sampling Methods
Graphs and Data Displays
Frequency Tables and Line Plots
Mean, Median, Mode, and Range
Stem-and-Leaf Plots
Measures of Variation
What is Probability?
Probability can help you solve all sorts of everyday problems, but first you need to
know what probability is! Follow along with this tutorial to learn about probability!
What are Compound Events?
What is Experimental Probability?
Do real life situations always work out the way your mathematical models tell you
they should? No! This tutorial describes how experimental probability differs from
theoretical probability.
How Do You Determine All the Possible Outcomes of an Experiment?
When you perform an experiment, how do you figure out all the possible
outcomes? Follow along with this tutorial to see!
What is the Fundamental Counting Principle?
The Fundamental Counting Principle is a way to figure out the total number of ways
different events can occur. In this tutorial, you'll be introduced to this principle and
see how to use it in an example. Take a look!
What is a Sample Space?
In an experiment, it's good to know your sample space. The sample space is the
set of all possible outcomes of an experiment. Watch this tutorial to get a look at
the sample space of an experiment!
What is an Outcome?
When you're conducting an experiment, the outcome is a very important part. The
outcome of an experiment is any possible result of the experiment. Learn about
outcomes by watching this tutorial!
How Do You Determine If Events are Independent or Dependent?
How Do You Find the Probability of Independent Events?
Calculating probabilities? Take a look at this tutorial and see how to figure out the
probability of independently drawing certain cards from a deck!
How Do You Find the Probability of a Simple Event?
Working with probabilities? Check out this tutorial! You'll see how to calculate the
probability of picking a certain marble out of a bag.
Nallari Math 8 Class
Probability Notes(INB)
March 24, 2016
Probability Notes
The probability of an event is a number describing the chance that the event will
happen. An event that is certain to happen has a probability of 1. An event that
cannot possibly happen has a probability of zero. If there is a chance that an
event will happen, then its probability is between zero and 1.
Examples of Events:
tossing a coin and it landing on heads
tossing a coin and it landing on tails
rolling a '3' on a die
rolling a number > 4 on a die
it rains two days in a row
drawing a card from the suit of clubs
guessing a certain number between 000 and 999 (lottery)
Events that are certain:
◦
If it is Thursday, the probability that tomorrow is Friday is certain,
therefore the probability is 1.
If you are sixteen, the probability of you turning seventeen on your next
birthday is 1. This is a certain event.
◦
Events that are uncertain:
◦
◦
The probability that tomorrow is Friday if today is Monday is 0.
The probability that you will be seventeen on your next birthday, if you
were just born is 0.
Let's take a closer look at tossing the coin. When you toss a coin, there are two
possible outcomes, "heads" or "tails." Examples of outcomes:
When rolling a die for a board game, the outcomes possible are 1,2,3,4,5, and 6.
The outcomes when choosing the days of a week are Sunday, Monday, Tuesday,
Wednesday, Thursday, Friday, and Saturday.
When selecting a door to be chosen behind on the game show "Let's make a Deal," the
outcome of the choices is door 1, door 2, and door 3.
The set of all possible outcomes of an experiment is the sample space for the
experiment.
A die is rolled and the number on the top face is recorded. All the integers from 1 to 6 are
possible. So S = {1, 2, 3, 4, 5, 6}.
A pair of dice is rolled and the sum of the numbers on the top faces is recorded. The smallest
sum is 1 + 1 = 2; the largest is 6 + 6 = 12; and all the integers in-between are possible. So S
= {2, 3, 4,..11, 12}.
The two outcomes of tossing a coin are equally likely, which means that each has
the same chance of happening. When all outcomes of an event are equally likely,
the probability that the event will happen is given by the ration below.
When looking at the probability of the event that the coin lands on tail we get the
following:
Continuation Next Page
Nallari Math 8 Class
Probability Notes(INB)
March 24, 2016
These events are equally likely to happen:
When there is a 50% chance of rain that means that there is a chance that it might
rain, but that there is also a chance that it might not rain. These chances are the
same so the event is equally likely to happen.
When one rolls a game die, he/she has exactly the same chance of landing on any
of the six sides. Therefore the probability of landing on any one specific side
would be 1/6. This is also true for any spinner. Say a spinner is divided into 10
sections. Then there is an equally likely chance that the spinner can land on any
of the sections. Thus the probability for the spinner to land in any designated
section is 1/10.
The probability of selecting one of the three doors on the game show is also
equally likely. There is no bias over the contestant’s decision so each door has a
probability of 1/3 being chosen. This is true if there are no arrows pointing
towards one of the doors.
When the outcome of one event does not affect the outcome of a second event,
the events are independent. When the outcome of one event does affect the
outcome of the second event, the events are dependent. Let's look at the
following five examples. Decide which pair of events are dependent or
independent.
1. Toss a coin. Then roll a number cube (die).
2. Choose a bracelet and put it on. Choose another bracelet.
3. Select a card. Do not replace it. Then select another card.
4. Select a card. Replace it. Select another card.
5. Pick one flower from a garden, then pick another.
The events above that are independent are numbers 1 and 4.
Probability of Two or Three Independent Events
If A, B and C are independent event, P(A, B and C)= P(A) X P(B) X P(C)
Given a coin and a die, what is the probability of tossing a head and rolling a 5?
The events above that are dependent events are numbers 2, 3, and 5.
Probability of Two or Three Dependent Events
Given a deck of cards, what is the probability that the second card drawn will be
from the same suit if the first card drawn is the Queen of Hearts?
The Counting Principle
The number of outcomes of an event is the product of the number of outcomes for each stage of the event.
Let's suppose that we want to find out how many different license plates are possible, when the license plate
is composed of three digits then three letters. To do this you can use the counting principle. The product
below gives the number of different plates possible.
We can use this same principle even if there are stipulations on the problem. For example say that a number
or letter can only be used once.
What if we wanted to know which plate system provided more possible outcomes, one with five letters or
one with seven digits?
Five letters: Seven digits: So we see by using the counting principle that there are more possibilities if we use five letters instead of
seven digits.
Nallari Math 8 Class