The probability that k number of vehicles arrive (i.e. cross

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The probability that k number of vehicles arrive (i.e. cross a predefined line) in time t is given as (λt)^ke^-λt/k! where λ is the average vehicle arrival rate. What is the probability that the time headway is greater than or equal to time t₁ ?

1. λe^λt₁
2. λe^{- t₁}
3. e^λt₁
4. e^{- λt₁}

Correct Option: D

Probability of k vehicles arrival,

P(k) =	(λt)^ke^-λt₁
	k!

Time headway ≥ t₁ means no vehicle arrives i.e., k = 0
∴ P(k) = e^-λt₁