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302-DECISION SCIENCE
Unit No.5. Probability
5.2.4 Case 1: Binomial
Probability Distribution
Presented By:
Dr. V. M. Tidake
Ph. D (Financial Management), MBA(FM), MBA(HRM) BE(Chem)
Dean EDP & Associate Professor MBA
2
Sanjivani College of Engineering, Kopargaon
Department of MBA
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Probability Distribution
10 Unbiased coins are tossed simultaneously, find the
probability that there will be-
i. Exactly 5 heads
ii. At least 8 heads
iii. Not more than 3 heads
iv. At least one head
v. If this exercise is carried out 50 times, how many times
we can get exactly 5 heads?
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Case 1: Binomial Distribution
Let, p - Probability of getting Head = Β½
Similarly, q - Probability of getting Tail = Β½
Also the experience is performed using 10 coins i.e. it is
carried out n = 10 number of times.
Hence the probability of getting r successes in 10 trials
is given by Binomial Probability Function as-
P(r) = 10Crprq10-r = 10Cr (
π
π
)r (
π
π
)10-r = 10Cr (
π
π
)10 = π
ππππ
10Cr
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Case 1: Binomial Distribution
i. Probability of getting 5 heads i.e. r = 5 is-
P(5) =
π
ππππ
10C5
P(5) =
π
ππππ
β
ππβπβπβπβπ
πβπβπβπβπ
P(5) = π. πππ
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Case 1: Binomial Distribution
ii. Probability of getting at least 8 heads i.e. r β₯ 8 is-
P(r β₯ 8) = P r = 8 or 9 or 10 = P 8 + P 9 + P 10
P(r β₯ 8) =
1
1024
(10C8 + 10C9 +10C10)
P(r β₯ 8) =
1
1024
(10C2 + 10C1 +10C0)
P(r β₯ 8) =
1
1024
β
10β9
1β2
+ 10 + 1
P(r β₯ 8) =
1
1024
β 56
P(r β₯ 8) = 0.055
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Case 1: Binomial Distribution
iii. Probability of not more than 3 heads i.e. r β€ 3 is-
P(r β€ 3) = P r = 0 or 1 or 2 or 3 = P 0 + P 1 + P 2 + P 3
P(r β€ 3) =
1
1024
(10C0 + 10C1 +10C2 + 10C3)
P(r β€ 3) =
1
1024
β 1 + 10 +
10β9
1β2
+
10β9β8
1β2β3
P(r β€ 3) =
1
1024
β 1 + 10 + 45 + 120 =
176
1024
P(r β€ 3) = 0.172
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Case 1: Binomial Distribution
iv. At least 1 head i.e. r β₯ 1 is-
P(r β₯ 1) = 1 β P r Λ 1 = 1 β P 0
P(r β₯ 1) = 1 -
1
1024
(10C0)
P(r β₯ 1) = 1 -
1
1024
P(r β₯ 1) =
1023
1024
P(r β₯ 1) = 0.9
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Case 1: Binomial Distribution
v. Possibility of getting 5 Heads:
If the exercise of throwing n=10 coins is repeated for
N=50 Times,
Then the number of times we expect to get 5 heads
is-
= f(5) = N * P(5)
= 50 * 0.246
=12.3 i.e. 12