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The Probability of Meeting a Car on a Deserted Highway

 

The probability of a car on a deserted highway for a 30 minute period is 0.95. What is the probability of its occurrence in 10 minutes?

 

This question is difficult to answer because the information that we have is not what we would like to have. However, in real life, it is a common situation.

 

We want to determine the probability relating to 10 minutes, with the probability relating to 30 minutes. We can not just divide 0.95 by three (although some people try to do this). Knowledge about the fact that the vehicle will occur during the 30 minutes does not help much, as this may happen at any time. The car can occur during the first 10-minute interval, or second, or third. During each of these periods, we can see two, or five, or a thousand cars, but still, it is considered as the appearance of the car.

 

What we really want to know is the probability of 0 cars during the 30-minute period. It’s pretty simple to find it. Since there is a chance of 95% that at least one vehicle will occur in 30 minutes, the probability that we won’t see a single car during this time interval must be equal to 0.05.

 

Three things should happen (or, conversely, should not happen) in order for 0 cars to appear during the 30-minute segment. Firstly, it should not be a single-vehicle during the first 10 minutes. Then, it should not be a single-vehicle during the second 10 minutes. Finally, it should not be a single-vehicle during the last 10 minutes. The question asks the probability of a car for a 10 minute period. Let’s call it x. The probability of the lack of cars in 10 minutes is equal to 1 – X. We multiply this value by itself three times. It should be equal to 0.05, that is:

 

(1 – x)^3 = 0.05 

Take a cube root from both parts:

 

1 – X = ^{3}\textrm{V}0.05 

Let’s solve this equation for x:

 

X = 1 – ^{3}\textrm{V}0.05 

Nobody expects you to find the cube roots in your head. The calculator will tell us that the answer is about 0.63. This result is well-founded. The likelihood of a car in the 10-minute period must be less than the probability of its occurrence in a 30-minute period which is equal to 0.95.

 

In this example, we see that even the most simple problem in the probability theory may be far from what it seems at first glance.

 

Our whole life submits to the laws of this science, so studying it is a very interesting and useful occupation.


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