Probability

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Unit 12 Probability
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Key Goals

• Understand and use tree diagrams to solve
  problems
• Compute the probability of outcomes when choices
  are equally likely
• Use the multiplication counting principle to find
  the total number of possible outcomes of a
  sequence of choices.
• Find the greatest common factor and least common
  multiple of two numbers
• Solve ratio and rate number stories
• Find the factors and prime factorizations of
  numbers

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Click me for help
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Understand and use tree diagrams to solve problems

• A tree diagram is a branched diagram that shows
  all possible choices.
• Tree diagrams are useful for solving probability
  problems when there are several options

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Tree diagrams can be a helpful way of organizing
outcomes in order to identify probabilities. For
example, if we have a box with two red, two green
and two white balls in it, and we choose two
balls without looking, what is the probability of
getting two balls of the same color? We use the
tree diagram to the left to help us identify the
possible combinations of outcomes. Here we see
that there are nine possible outcomes, listed to
the right of the tree diagram. This number is the
size of the sample space for this two step
experiment, and will be in the denominator of
each of our probabilities. Each of these
possible nine outcomes has a probability of
1/9. Because there are three instances out of the
nine that result in 2 balls of the same
color P(2 the same color) 3/9 (This can be
reduced to 1/3)
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Try it out!
What is the probability of getting a white ball
in the second stage?
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Try it out!
What is the probability of getting a white ball
in the second stage?
The probability is 1/3 because of the 9
possibilities, there are 3 white balls. 3/9 1/3
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Try it out!
What is the probability of getting a white and a
green ball in no particular order?
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Try it out!
What is the probability of getting a white ball
in the second stage?
The probability is 2/9 because of the 9
possibilities, there are 2 combinations with
those colors- WG and GW If the order did matter,
then the probability would be 1/9 because there
is a 1/3 chance of the first balls color being
white and a 1/3 chance of the second ball being
green. 1/3 x 1/3 1/9
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Compute the probability of outcomes when choices
are equally likely.
Probability can be determined by giving the total
number of desired outcomes over the total number
of possibilities.
For example, on a six-sided die, there is a 1/6
chance of rolling a 4.
If you wanted to find the chances of rolling a 2
or a 4, you would add the probability of getting
a 2 to the probability of getting a 4. If you
wanted to know the probability of getting a 2 and
then a 4 (order matters), then you would multiply
the probabilities together. 1/6 x 1/6 1/12
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Try it out!
Read about probability and take a quiz Play
probability games Crazy choices
game Theoretical vs. Experimental probability
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Use the multiplication counting principle to find
the total number of possible outcomes of a
sequence of choices.

• When you have a sequence events, the likelihood
  of the events happening together in a particular
  order is less than the probability of them
  happening separately or in no particular order.
• When you have dependent events, you will
  multiply the probabilities together to calculate
  the probability of both events happening.
• - You do this because probability is expressed as
  a fraction. If you add two fractions together,
  the sum will be greater than each fraction. When
  you multiply two fractions together, the product
  is smaller than the first fraction because you
  are finding a part of a part.

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Try It Out!
Click here to print a probability activity. Use
the dice and the coins on this page to help
answer the questions. Turn your work in to Mrs.
R. when you finish.
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Name _______________
Use the links on the previous slide to help you
complete this chart. Give the theoretical and
experimental probability for each event described
and analyze and explain the relationship between
your theoretical and experimental probability.
Remember that theoretical probability is based
only on math. Experimental probability is based
on actual experimenting.
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Find the Greatest Common Factor and Least Common
Multiple of two numbers
The Greatest Common Factor (GCF) is the largest
number that two numbers are both divisible by.
To find the GCF of two numbers List the prime
factors of each number. Multiply those factors
both numbers have in common. If there are no
common prime factors, the GCF is 1.
Explanation and Game
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Find the Greatest Common Factor and Least Common
Multiple of two numbers
The Least Common Multiple (LCM) is the smallest
multiple that both numbers have in common.
A common multiple is a number that is a multiple
of two or more numbers. The common multiples of 3
and 4 are 0, 12, 24, .... The least common
multiple (LCM) of two numbers is the smallest
number (not zero) that is a multiple of both.
Explanation and Game
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Try It Out!
Which is the LCM of 6 and 10?
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Try It Out!
Which is the LCM of 6 and 10?
The LCM of 6 and 10 is 30 If you list the
multiples of 6, you have 6, 12, 18, 24, 30 If
you count by 10s, the first one of those numbers
you come to is 30.
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Try It Out!
Which is the LCM of 12 and 8?
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Try It Out!
Which is the LCM of 12 and 8?
The LCM of 12 and 8 is 24 If you list the
multiples of 8, you have 8, 16, 24, 32 If you
list the multiples of 12, the first of those
numbers you come to is 24.
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Try It Out!
Which is the GCF of 12 and 8?
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Try It Out!
Which is the GCF of 12 and 8?
The GCF of 12 and 8 is 4 If you list the factors
of 8, you have 1, 2, 4, 8 If you list the
factors of 12, you have 1, 2, 3, 4, 6, 12 4 is
the largest factor they have in common.
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Try It Out!
Which is the GCF of 24 and 38?
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Try It Out!
Which is the GCF of 24 and 38?
The GCF of 24 and 38 is 2 If you list the
factors of 38, you have 1, 2, 19, 38 If you
list the factors of 24, you have 1, 2, 3, 4, 6,
8, 12, 24 2 is the largest factor they have in
common.
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Solve Ratio and Rate Number Stories
A ratio is a way of comparing amounts of
something. It shows how much bigger one thing is
than another. For example The ratio of
footballs to soccer balls is 43 This can also be
written as 4 to 3 or 4/3
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A rate is a ratio that compares two different
kinds of numbers, such as miles per hour or
dollars per pound. A unit rate compares a
quantity to its unit of measure. A unit price
is a rate comparing the price of an item to its
unit of measure. The rate "miles per hour" gives
distance traveled per unit of time. Problems
using this type of rate can be solved using a
proportion, or a formula.
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Rate is a very important type of ratio, used in
many everyday problems, such as grocery shopping,
traveling, medicine--in fact, almost every
activity involves some type of rate. Miles per
hour or feet per second are both rates of speed.
Number of heartbeats per minute is called "heart
rate." If you ask a babysitter, "What is your
rate?", you are asking how many dollars per hour
you will be charged. The little word "per" is
always a clue that you are dealing with a rate.
Unit price is a particular rate that compares a
price to some unit of measure. For example,
suppose eggs are on sale for .72 per dozen. The
unit price is .72 divided by 12, or 6 cents per
egg. The word "per" can be replaced by the "/"
in problems, so 6 cents per egg can also be
written 6 cents/egg.
Click me to try some practice problems!
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Try It Out!
13
2
26
15
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Try It Out!
The correct answer is 13 because For every free
throw, they attempt two field goals. If they make
13 free throws, they would attempt 26 field
goals. If they attempt 26 field goals, they will
make half of them for a total of 13 expected
baskets.
13
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Try It Out!
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18
7
11
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Try It Out!
The correct answer is 18 because 4 wins 7
losses total matches 11 4 wins 7 losses total
matches 22 4 wins 7 losses total matches 33 4
wins 7 losses total matches 44 2 wins 4
losses total matches 50 18 total wins
18
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Try It Out!
If the tortoise moves 3 feet per minute, and the
hare moves 9 feet per minute, how long will it
take each to reach the finish line?
27 feet
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Try It Out!
If the tortoise moves 3 feet per minute, and the
hare moves 9 feet per minute, how long will it
take each to reach the finish line?
27 feet / 9 feet 3 minutes
27 feet / 3 feet 9 minutes
27 feet