Puzzle – Probability of Seeing a Car in 10 Minutes & 30 Minutes

Question:

In thirty minutes on a deserted road, there is a 95% chance of seeing a vehicle. What are the probabilities of seeing a vehicle in ten minutes?

Solution:

To make the question more understandable, we can state that the probability of seeing any other vehicle in the next 30 minutes is 95%, or, more precisely and practically, the probability of not seeing any other vehicle is 5%. 

So, we need to find 3 consecutive times without seeing another vehicle for 10 minutes for it to be 30 minutes before we see another one. 

Let the probability of not seeing a vehicle in 10 minutes be P(not10) = P

Therefore, the probability of not seeing a vehicle in 30 minutes, 

P(not30) = P(not10)*P(not10)*P(not10)= P*P*P = P^3

Given that, there is a 95% chance of seeing a vehicle (i.e., 0.95) in 30 minutes:

We can find the probability of seeing any other vehicle in the next 30 minutes is, 

            = 1 – P(not30) = 0.95 

            = 1 – P^3 = 0.95

            =  P^3 = 1 – 0.95 = 0.05

           P = 3√0.05 = 0.3684

Therefore, the probability of seeing a vehicle in the next 30 minutes is  P = 0.3684

Therefore, the probability of seeing a vehicle in 10 minutes,

 = 1- P = 1 – 0.3684 = 0.6315 

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