Given that a car travelling east at a certain speed passes two police stations a certain distance apart, one at time T and the other at time T+2, what is the probability that the car was at exactly the halfway point between the two police stations at time T+1?
To find the probability that the car was at the halfway point between the two police stations at time T+1, we need to understand the motion of the car and the time intervals involved.
Let's represent the car's speed as "v" (in units of distance per time) and the distance between the two police stations as "d" (in units of distance). We can assume that the car maintains a constant speed throughout its journey.
Given that the car passed the first police station at time T and the second police station at time T+2, we can deduce that the time interval between the two police stations is 2 units of time.
To calculate the distance covered by the car during this time interval, we multiply the car's speed "v" by the time interval "2". Therefore, the distance covered by the car is 2v.
Since the car is traveling at a constant speed, we can assume that it covers equal distances in equal time intervals. Therefore, the distance between the car and the first police station at time T+1 is v, and the distance between the car and the second police station at time T+1 is also v.
If the car is exactly at the halfway point between the two police stations at time T+1, then the distance between the car and each police station at that time should be equal.
Hence, the probability that the car was at exactly the halfway point between the two police stations at time T+1 is equal to the probability that the car was at a distance of v from each police station at that time.
Since the car's position is continuous and can take any real value within the interval, the probability is essentially the ratio of the length of the interval in which the car's position would make it halfway between the two police stations at time T+1 to the total length of the possible positions.
To calculate this probability, we need more information about the car's speed, the distance between the police stations, or any constraints on the possible positions of the car. Without this additional information, it is not possible to determine the exact probability.
To find the probability that the car was at the halfway point between the two police stations at time T+1, we need to consider the possible positions and movements of the car.
Let's assume that the car's speed is constant and that it travels in a straight line between the two police stations.
To simplify the problem, let's consider the distance between the two police stations as a unit of measurement, where the distance is 1 unit.
Case 1: The car is at the halfway point between the two police stations at time T.
In this case, the car starts at the T-1 position and reaches the halfway point at time T.
The probability of this specific event occurring is 1, as it is given in the problem.
Case 2: The car is at the halfway point between the two police stations at time T+2.
In this case, the car starts at the T+1 position and reaches the halfway point at time T+2.
The probability of this specific event occurring is also given as 1 because it is stated in the problem.
Since there are only two possible cases in which the car can be at the halfway point (at time T or at time T+2) and both have a probability of 1, the total probability is the sum of these probabilities. Thus, the probability that the car was exactly at the halfway point between the two police stations at time T+1 is:
P = 1 + 1 = 2
Therefore, the probability is 2.