12'6 Binomial Distribution

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Permutation
Notation
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Permutation
When deciding who goes 1st, 2nd, etc., order is
important.
A permutation  is an arrangement or listing of
objects in a specific order. 
The order of the arrangement is very important!! 
The notation for a permutation       nPr
n  is the total number of
objects
r is the number of objects selected (wanted)
Note  if  n r   then   nPr    n!
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Permutations

• Simplify each expression.
• a. 12P2
• b. 10P4
• c. At a school science fair, ribbons are given
  for first, second, third, and fourth place, There
  are 20 exhibits in the fair. How many different
  arrangements of four winning exhibits are
  possible?

12 11 132
10 9 8 7 5,040
20P4 20 19 18 17 116,280
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Combinations

• A selection of objects in which order is not
  important.
• Example 8 people pair up to do an assignment.
  How many different pairs are there?

5
Combinations

• AB AC AD AE AF AG AH
• BA BC BD BE BF BG BH
• CA CB CD CE CF CG CH
• DA DB DC DE DF DG DH
• EA EB EC ED EF EG EH
• FA FB FC FD FE FG FH
• GA GB GC GD GE GF GH
• HA HB HC HD HE HF HG

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Combinations

• The number of r-combinations of a set with n
  elements,
• where n is a positive integer and
• r is an integer with 0 lt r lt n,
• i.e. the number of combinations of r objects from
  n unlike objects is

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Example 1

• How many different ways are there to select two
  class representatives from a class of 20
  students?

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Binomial Distribution

• p.212 in the math 3 book

9
Many Experiments can be done with the results of
each trial reduced to 2 outcomes

• Binomial Experiment
• There are n independent trials
• Each trial has only 2 possible outcomes
• Success or failure.
• The probability of success is the same for each
  trial.
• The probability is p.
• The probability of failure is 1-p

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Finding a Binomial Probability

• For a binomial experiment consisting of n trials,
  the probability of exactly K successes is
• P(k successes) nCk pk (1-p)n-k
• Where the probability of success on each trial is
  p.

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EXAMPLE 1

• At a college, 53 of students receive financial
  aid. In a random group of 9 students, what is the
  probability that exactly 5 of them receive
  financial aid?
• p.53 (the prob of success for each trial)
• n9 (diff trials or experiments)
• The prob of getting 5 successes (k5)
• P(k5) 9C5 .535 (1-.53)9-5
• about 26

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EXAMPLE 2

• Draw a bar graph of the binomial distribution for
  the class of students in Example 1 and find the
  probability that fewer than 3 students in the
  class receive financial aid.
• Hint Use P(k successes) nCk pk (1-p)n-k

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• P(k0) 9C0 .530 (1-.53)9-0 .001
• P(k1) 9C1 .531 (1-.53)9-1 .011
• P(k2) 9C2 .532 (1-.53)9-2 .05
• P(k3) 9C3 .533 (1-.53)9-3 .13
• P(k4) 9C4 .534 (1-.53)9-4 .23
• P(k5) 9C5 .535 (1-.53)9-5 .26
• P(k6) 9C6 .536 (1-.53)9-6 .19
• P(k7) 9C7 .537 (1-.53)9-7 .09
• P(k8) 9C8 .538 (1-.53)9-8 .03
• P(k9) 9C9 .539 (1-.53)9-9 .003

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(No Transcript)
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• Probability of fewer than 3
• P(0) P(1) P(2)
• .001 .011 .05
• 0.062

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EXAMPLE 4

• A diet company claims that 78
• of people who use their product
• lose weight. To test this hypothesis,
• you randomly choose 9 people
• and ask them to follow the
• diet for 3 months. Of the 9 people,
• only 4 people lose weight. Should
• you reject the claim? Explain your
• reasoning.
• P(k 4) 0.02 which
• is fairly small, so you should reject
• the claim.

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The binomial distribution describes the behavior
of a count variable X if the following conditions
apply 1 The number of observations n is
fixed. 2 Each observation is independent. 3
Each observation represents one of two outcomes
("success" or "failure"). 4 The probability of
"success" p is the same for each outcome. If
these conditions are met, then X has a binomial
distribution with parameters n and p, abbreviated
B(n,p).
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Assignment

• Page 216 math 3 book
• s 1 18
• Read over page 217 (try to figure it out on the
  calculatorstill stuck? Well go over).
• s 1-3