4)Suppose the probability that a bit transmitted through a digital communication
channel and received in error is 0.01. Assuming that the transmissions are
independent events,
a) Find the probability that the first error occurs at the 8th bit.
b) Find the probability that the third error occurs at the 8th bit.
c) Find the probability that three or more errors occur among 100 transmitted bits.
Answers
Given : probability that a bit transmitted through a digital communication
channel and received in error is 0.01.
To find : probability that the first error occurs at the 8th bit. probability that the third error occurs at the 8th bit. probability that three or more errors occur among 100 transmitted bits.
Solution:
probability of Error p = 0.01
probability of no error = q = 1 - p = 0.99
P(x) = ⁿCₓpˣqⁿ⁻ˣ
probability that the first error occurs at the 8th bit.
=> no Error in 7 bits * error in 8th
Probability = (no error in 7 ) ( error in 8th)
(⁷C₀(0.01)⁰(0.99)⁷ ) (0.01) = 0.00932
Find the probability that the third error occurs at the 8th bit.
2 Error Occurred in 7 & 3rd in 8th
Probability = (2 error in 7 ) ( error in 8th)
(⁷C₂(0.01)²(0.99)⁵ ) (0.01) = 0.00001997 = 0.00002
probability that three or more errors occur among 100 transmitted bits.
= 1 - 0 error - 1 error - 2 Error
= 1 - ¹⁰⁰C₀(0.01)⁰(0.99)¹⁰⁰ - ¹⁰⁰C₁(0.01)¹(0.99)⁹⁹ - ¹⁰⁰C₂(0.01)²(0.99)⁹⁸
= 0.07937
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