Transcript 9th and 10th Grade PD Session 3 - southmathpd

Data Analysis and Probability
Presenters
Aaron Brittain
Adem Meta
 MA.9.5.H: Use counting techniques, such as
permutations and combinations, to determine the
total number of options and possible outcomes.
 MA.9.5.I: Design an experiment to test a
theoretical probability, and record and explain
results.
 MA.9.5.J: Compute probabilities of compound
events, independent events, and simple
dependent events.
 MA.9.5.K: Make predictions based on theoretical
probabilities and experimental results.
 INDEPENDENT EVENTS: when the
occurrence of one event has no
effect on the probability that a
second event will occur.
 DEPENDENT EVENTS: is the
occurrence of one event does have
an effect on the probability that a
second event will occur.
 PROBABILITY OF TWO INDEPENDENT EVENTS
 P(A and B) = P(A) x P(B)
 EX: Find the probability of flipping a coin and
getting heads and then rolling a 6 on a number
cubed.
 1. Identify the event as independent or
dependent.
 2. Find the probability of each event
 P(heads) = ½
 P(6 on a number cube) = 1/6
 3. multiply the two results and
simplify your fraction if necessary
 ½ x 1/6 = 1/12
 Holt 10.7 Additional Algebra Lab
 Pp. 734-735
 MA.9.5.J: Compute probabilities of compound events,
independent events, and simple dependent events.
 Holt 10.8 Algebra Lab
 Pp. 736-743
 MA.9.5.H: Use counting techniques, such as permutations and
combinations, to determine the total number of options and
possible outcomes.
 Combinations and Permutations
 What's the Difference?
 In English we use the word "combination" loosely, without thinking if the
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order of things is important. In other words:
"My fruit salad is a combination of apples, grapes and bananas" We
don't care what order the fruits are in, they could also be "bananas,
grapes and apples" or "grapes, apples and bananas", its the same fruit
salad. "The combination to the safe was 472". Now we do care about
the order. "724" would not work, nor would "247". It has to be exactly 47-2.
So, in Mathematics we use more precise language:
If the order doesn't matter, it is a Combination. If the order does matter
it is a Permutation.
So, we should really call this a "Permutation Lock"!
In other words:
 A Permutation is an ordered Combination.
 To help you to remember, think "Permutation ...
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Position"
Permutations
There are basically two types of permutation:
Repetition is Allowed: such as the lock above. It could be
"333".
No Repetition: for example the first three people in a
running race. You can't be first and second.
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1. Permutations with Repetition
These are the easiest to calculate.
When you have n things to choose from ... you have n choices each time!
When choosing r of them, the permutations are:
n × n × ... (r times)
(In other words, there are n possibilities for the first choice, THEN there
are n possibilities for the second choice, and so on, multplying each time.)
Which is easier to write down using an exponent of r:
n × n × ... (r times) = nPr
Example: in the lock above, there are 10 numbers to choose from (0,1,..9)
and you choose 3 of them:
10 × 10 × ... (3 times) = 10^3 = 1,000 permutations
So, the formula is simply:
nPr where n is the number of things to choose from, and you choose r of
them
(Repetition allowed, order matters)
 2. Permutations without Repetition
 In this case, you have to reduce the number of available choices
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each time.
For example, what order could 16 pool balls be in?
After choosing, say, number "14" you can't choose it again.
So, your first choice would have 16 possibilites, and your next choice
would then have 15 possibilities, then 14, 13, etc. And the total
permutations would be:
16 × 15 × 14 × 13 × ... = 20,922,789,888,000
But maybe you don't want to choose them all, just 3 of them, so
that would be only:
16 × 15 × 14 = 3,360
Combinations and Permutations
 One class has 20 students. If the math teacher wants to
create groups of 3 students, how many possibilities are
there?
 In the same class, if they were to elect in each group one
president, one vice-president, and one secretary, how
many possible groups are there?
Combinations and Permutations
 Ohio license plates have four numbers and three letters.
How many plates can be produced?
 If a group of five people are in the room at the same time,
how many handshakes are there if the all greet each
other?
Binomial Distributions
 Probability Mass Function The binomial distribution is
used when there are exactly two mutually exclusive
outcomes of a trial. These outcomes are appropriately
labeled "success" and "failure". The binomial distribution
is used to obtain the probability of observing x successes
in N trials, with the probability of success on a single trial
denoted by p. The binomial distribution assumes that p is
fixed for all trials.
Binomial Distributions
 The formula for the binomial
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probability mass function is:
P( x, n, p)  nCx( p) (1  p)
x
n x
Binomial Distributions
 On the 2009 OGT Math Test, there were 31 multiple
choice questions. (4 possible answers for each question)
 What is the probability that a student to get 20 correct
answers by random selection ?
 What is the probability to get at least 27 correct answers?
 What is the probability to get at most 4 correct answers?
Binomial Distributions
 If a person toss a coin three times in a row, find the
number possibilities in the sample space. (draw the tree)
 What is the probability to get three tails?
 What is the probability to get at least two heads?
 What is the probability to get at most one tail?
Sample Space
 Create a sample space for the total number of
outcomes when rolling two number cubes
Find the probability of the sum of two numbers
equaling six
 MA.9.5.H: Use counting techniques, such as permutations
and combinations, to determine the total number of
options and possible outcomes.
 Thank You for Attending
 Please be sure to check out
 cmsdmathcoaches.pbworks.com