binomial lecture
DESCRIPTION
Binomial LectureTRANSCRIPT
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The Binomial Experiment
• Repeated n times(trials) under identical
conditions
• Each trial can result in only one out of two
outcomes
– Success – probability success p
– Failure – probability failure q = 1 – p
• Trials are independent
• Measure number of successes, x, in n trails
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The Binomial Experiment
Typical cases where the binomial experiment
applies:
– A coin flipped results in heads or tails
– A party wins or loses election
– An employee is male or female
– A car uses leaded, or unleaded fuel
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The Binomial Experiment
• Binomial distribution is the probability distribution
that applies to the binomial experiment
• Displayed in the form of a table where the first
row (or column) displays all possible number of
successes, second row (or column) displays the
probability associated with number of successes
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-( ) ( ) x n x
n xP X x P x C p q
Determining x successes in n trials:
where, = number of trails = probability of a success = probability of a failure = number of successes
!
!( - )!n x
npqx
nC
x n x
Calculating the Binomial Probability
The Binomial Experiment
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• Repeated n = 5 times
• Each trial can result in only one out of two outcomes – Success – late for class → p = 0.10
– Failure – not late for class → q = 1 - 0.10 = 0.90
• Students are independent
Are the conditions required for the binomial experiment met?
• 10% of students are late for the early morning class
• In a sample of 5 students, find the probability distribution of the number students that are late
The Binomial Experiment - Example
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• Let X be the binomial random variable indicating
the number of late students
0 5-0
5 0P(X = 0) = P(0) = C (0.10) (0.90) = 0.5905
5 5-5
5 5P(X = 5) = P(5) = C (0.10) (0.90) = 0.00001
The Binomial Experiment - Example
Calculate the probability that zero students are late Calculate the probability that one student is late Calculate the probability that three students are late
1 5-1
5 1P(X = 1) = P(1) = C (0.10) (0.90) = 0.3281
3 5-3
5 3P(X = 3) = P(3) = C (0.10) (0.90) = 0.008
2 5-2
5 2P(X = 2) = P(2) = C (0.10) (0.90) = 0.072
4 5-4
5 4P(X = 4) = P(4) = C (0.10) (0.90) = 0.00045
-( ) ( ) (1 - )x n x
n xP X x p x C p p
• Let X be the binomial random variable indicating
the number of late students
The Binomial Experiment - Example
0 5-0
5 0P(X = 0) = P(0) = C (0.10) (0.90) = 0.5905
5 5-5
5 5P(X = 5) = P(5) = C (0.10) (0.90) = 0.00001
1 5-1
5 1P(X = 1) = P(1) = C (0.10) (0.90) = 0.3281
3 5-3
5 3P(X = 3) = P(3) = C (0.10) (0.90) = 0.008
2 5-2
5 2P(X = 2) = P(2) = C (0.10) (0.90) = 0.072
4 5-4
5 4P(X = 4) = P(4) = C (0.10) (0.90) = 0.00045
X P(X)
0 0.5905
1 0.3281
2 0.0729
3 0.0081
4 0.00045
5 0.00001
∑P(X) ≈ 1
• Calculate the probability that 2 or less students
will be late
The Binomial Experiment - Example
X P(X)
0 0.5905
1 0.3281
2 0.0729
3 0.0081
4 0.00045
5 0.00001
∑P(X) ≈ 1
P(X ≤ 2)
= P(X = 0) + P(X = 1) + P(X = 2)
= 0.5905 + 0.3281 + 0.0729
= 0.9915
The Binomial Experiment - Example
X P(X)
0 0.5905
1 0.3281
2 0.0729
3 0.0081
4 0.00045
5 0.00001
∑P(X) ≈ 1
P(X < 2)
= P(X = 0) + P(X = 1)
= 0.5905 + 0.3281
= 0.9186
• Calculate the probability that less than 2 students
will be late
The Binomial Experiment - Example
X P(X)
0 0.5905
1 0.3281
2 0.0729
3 0.0081
4 0.00045
5 0.00001
∑P(X) ≈ 1
P(X ≥ 4)
= P(X = 4) + P(X = 5)
= 0.00045 + 0.00001
= 0.00046
• Calculate the probability that 4 or more than 4
students will be late
The Binomial Experiment - Example
X P(X)
0 0.5905
1 0.3281
2 0.0729
3 0.0081
4 0.00045
5 0.00001
∑P(X) ≈ 1
P(X > 4)
= P(X = 5)
= 0.00001
• Calculate the probability that more than 4
students will be late
The Binomial Experiment - Example
X P(X)
0 0.5905
1 0.3281
2 0.0729
3 0.0081
4 0.00045
5 0.00001
∑P(X) ≈ 1
P(X ≥ 3)
= P(X = 3) + P(X = 4) + P(X = 5)
= 0.00856 OR
= 1 – P(X ≤ 2)
= 1 – 0.9915
= 0.0085
• Calculate the probability that 3 or more students
will be late
The Binomial Experiment - Example
X P(X)
0 0.5905
1 0.3281
2 0.0729
3 0.0081
4 0.00045
5 0.00001
∑P(X) ≈ 1
P(X > 3)
= P(X = 4) + P(X = 5)
= 0.00046 OR
= 1 – P(X ≤ 3)
= 1 – 0.9996
= 0.0004
• Calculate the probability that more than 3 students
will be late
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• Mean and standard deviation of binomial
random variable
The Binomial Experiment
( )E X np 2 ( )Var X npq
REMEMBER
Repeated n times(trials) under identical conditions
Each trial can result in only one out of two outcomes
Success – probability success p
Failure – probability failure q = 1 – p
Measure number of successes, x, in n trails
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– What is the expected number of students that come
late?
– What is the standard deviation for the number of
students who come late?
( ) 5(0.10) 0.5E X np
2 ( ) 5(0.10)(0.90) 0.67Var X npq
The Binomial Experiment - Example