Very important Discrete Probability Distribution

1. Binomial Distribution
Very important Discrete Probability Distribution
Dr. Anjali Devi JS
Guest Faculty
School of Chemical Sciences
M G University
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2. Combination
nCr=
𝑛!
𝑟! 𝑛−𝑟 !
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3. Consider tossing of a coin, what are the possible outcomes?
. Head or Tail
Consider tossing of a coin 5 times, how many total outcomes?
. 25=32
Probability of obtaining no head
HHHHH, THHHH….
P(X=0)=
1
32
=
5C0
32
5C0=
5!
0! 5−0 !
=1
=
5!
5!
Probability of obtaining one head
P(X=1)=
HTTTT, THTTT….
5
32
=
5C1
32
P(X=2)? =
5C2
32 P(X=3) =
5C3
32
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Binomial Experiments

4. Event Formula for probability Probability
Obtaining no head
P(X=0)
5C0
32
1
32
Obtaining 1 head P(X=1) 5C1
32
5
32
Obtaining 2 head P(X=2) 5C2
32
10
32
Obtaining 3 head P(X=3) 5C3
32
10
32
Obtaining 4 head P(X=4) 5C4
32
5
32
Obtaining 5 head P(X=5) 5C5
32
1
32
Probability of obtaining head in 5 trials
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5. X=0 X=1 X=2 X=3 X=4 X=5
1/32
5/32
10/32 10/32
5/32
1/32
Visualising Binomial Distribution
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6. What is a Binomial Distribution
 The number of success in n independent Bernoulli trials has a
binomial distribution.
Suppose:
There are n independent trials.
Each trial can result in one of two possible outcomes, labelled
SUCCESS and FAILURE
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7. What is a Binomial Distribution
P(Success) =p
P(Failure) =1-p
X represents the number of success in n trials
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8. Binomial Distribution Formula
Then X has a binomial distribution.
P(X=x) =nCr px (1-p)(n-x)
for x=0,1,2,3…
X= total number of success (head or tail, pass or fail)
P=Probability of success on an individual trial
n= number of trials
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9. Binomial distribution can be thought of as simply the probability of
SUCCESS or FAILURE outcome in an experiment that is repeated
multiple times.
The binomial is a type of distribution that has two possible
outcomes. The prefix bi means two or twice.
Binomial Distribution
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10. Criteria of Binomial Distribution
 The number of observation or trial is fixed.
Toss a coin:
P(Head)=50% P(Tail)=1-50=50%
 Each observation or trial is independent.
None of the trial have an effect on the
probability of next trial
 The probability of success (tail or head, fail or pass) is
exactly same from one trial to another 10

11. Notation
The random variable X
Is distributed
Using Binomial distribution with n
trials and probability P of specific
outcome
Example Suppose X is a binomial random variable with n=20 and P=0.5,
then notation:
𝑋~𝐵(20,0.5)
If a random variable X is binomially distributed (i.e., its probability function
uses a binomial distribution), then we write:
𝑋~𝐵(𝑛, 𝑃)
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12. Question
A coin is tossed 10 times. What is the probability of getting exactly 6 heads
P(X=x) =nCr px (1-p)(n-x)
for x=0,1,2,3…
P( Head) =0.5 1-P =0.5 x =6
P(X=6) =10C6 (0.5)6 (0.5)(4)
Answer
0.20 12

13. Question
A coin is tossed 5 times. What is the probability of getting exactly one head
P(X=x) =nCr px (1-p)(n-x)
for x=0,1,2,3…
P( Head) =0.5 1-P =0.5 x =1
Answer
P(X=1) =5C1 (0.5) 1 (0.5)(4)
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14. Variance = nP(1-P)
Mean = nP
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Variance & Mean- Binomial Distribution
Where n is number of trials and P is probability of specific
outcome.

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Thank You