pascal’s triangle - a triangular arrangement of where each row corresponds to a value of n. vocab...
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Pascal’s triangle - A triangular arrangement of where each row corresponds to a value of n.
VOCAB REVIEW:
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Pascal’s Triangle
This is a Pascal’s Triangle – Each row is labeled as n
• The first row is n = 0• Second is n = 1 etc…
– Each term is nCr • n is the row• r is the position in the row starting with 0
– Each term is the sum of the two directly above it.
n = 0; 20
n = 1; 21
n = 2; 22
n = 3; 23
n = 4; 24
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1. The 6 members of a Model UN student club must choose 2
representatives to attend a state convention. Use Pascal’s
triangle to find the number of combinations of 2 members that
can be chosen as representatives.
2. Use Pascal’s triangle again
to find the number of combinations
of 2 members that can be chosen
if the Model UN club has 7
members.
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The Binomial Theorem• Gives us the coefficients for a binomial expansion
• The values in a row of Pascal's triangle are the coefficients in a binomial expansion of the same degree as the row.
• A binomial expansion of degree n is (a + b)n.
• The variables are
anb0 + an-1b1 + … + a1bn-1 + anb0 + a0bn
Binomial Theorem
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32x y
1st term
2nd term
Pascal’s #
3 2 1 02 2 2 2 x x x x
0 1 2 3 y y y y
1 3 3 1 6 4 2 2 33 3 x x y x y y
6 4 2 1 x x x
2 31 y y y
Expand a Power of a Binomial Sum
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42a b
1st term
2nd term
Pascal’s #
4 3 2 1 0 a a a a a
0 1 2 3 42 2 2 2 2b b b b b
1 4 6 4 1
4 3 2 2 3 48 24 32 16a a b a b ab b
4 3 2 1 a a a a
2 3 41 2 4 8 16b b b b - -
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Use the binomial theorem to write the binomial expansion.
1.
2.
53x
42 p q
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3.
4.
42a b
35 2y
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• Find the coefficient of in the expansion of
where rm/p
(1 ) (2 ) r n rn rC st nd
np qax by
mx
Find a Coefficient in an Expansion
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• Find the coefficient of x⁴ in the expansion of (3x + 2)¹º.
n =
r =
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1. Use the binomial formula to find the coefficient of the term in the expansion of
2. Find the coefficient of the x5 in the expansion of (x – 3)7?
3. Find the coefficient of the x3 in the expansion of (2x +5)8?
9q z 10
3q z
Binomial Formula
5x
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12.3 An Introduction to Probability
What do you know about probability?• Probability is a number from 0 to 1 that
tells you how likely something is to happen.
• Probability can have two main approaches -experimental probability
-theoretical probability
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Experimental vs.Theoretical
Experimental probability:
P(event) = number of times event occurs
total number of trials
Theoretical probability:
P(E) = number of favorable outcomes total number of possible outcomes
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How can you tell which is experimental and which is theoretical probability?
Experimental:
You tossed a coin 10 times and recorded a head 3 times, a tail 7 times
P(head)= 3/10
P(tail) = 7/10
Theoretical:
Toss a coin and getting a head or a tail is 1/2.
P(head) = 1/2
P(tail) = 1/2
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Experimental probability
Experimental probability is found by repeating an experiment and observing the outcomes.
P(head)= 3/10
A head shows up 3 times out of 10 trials,
P(tail) = 7/10
A tail shows up 7 times out of 10 trials
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Theoretical probability
P(head) = 1/2P(tail) = 1/2Since there are only
two outcomes, you have 50/50 chance to get a head or a tail.
HEADS
TAILS
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How come I never get a theoretical value in both experiments? Tom asked.
• If you repeat the experiment many times, the results will getting closer to the theoretical value.
• Law of the Large Numbers
Experimental VS. Theoretical
50
53.4
48.948.4
49.87
45
46
47
48
49
50
51
52
53
54
1
Thoeretical5-trial10-trial20-trial30-trial
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Law of the Large Numbers 101
• The Law of Large Numbers was first published in 1713 by Jocob Bernoulli.
• It is a fundamental concept for probability and statistic.
• This Law states that as the number of trials increase, the experimental probability will get closer and closer to the theoretical probability.
http://en.wikipedia.org/wiki/Law_of_large_numbers
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Contrast experimental and theoretical probability
Experimental probability is the result of an experiment.
Theoretical probability is what is expected to happen.
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You must show the probability set up, the unreduced fraction,and the reduced fraction in order to receive full credit.
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Geometric Probability
• Geometric probabilities are found by calculating a ratio of two side lengths, areas, or volumes according to the problem.
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Find a Geometric Probability
• You throw a dart at the square board. Your dart is equally likely to hit any point inside the board. Are you more likely to get 10 points or 0? (use area)
2 5 10
0
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• HW 37: pg 719, 13-43 odd