Q: A train and a bus arrive at random between 9AM and 10AM.
The train stops for 10 minutes and bus for ‘x’ minutes. Find
‘x’ so that the bus and train will meet with probability of 0.5
A train and a bus arrive at the station at random between 9 A.M. and 10 A.M. The train stops
for 10 minutes and the bus for x minutes. Find x so that the probability that the bus and the
train will meet equals 0.5.
To find the value of 'x' such that the bus and train will meet with a probability of 0.5, we need to determine the duration of time during which they can potentially meet.
Let's analyze the situation:
1. The train arrives at a random time between 9AM and 10AM, so it has a total time span of 60 minutes.
2. The bus also arrives at a random time between 9AM and 10AM, so it also has a total time span of 60 minutes.
3. The train stops for exactly 10 minutes at its arrival time.
Now, let's consider the two scenarios in which the bus and train can potentially meet:
Scenario 1: The bus arrives before the train:
In this case, the bus has a time window of 60 minutes to wait for the train to arrive and stop for 10 minutes. Therefore, the bus can meet the train during the time window of 0 to 50 minutes past the hour. If the train arrives within this time window, they will meet.
Scenario 2: The train arrives before the bus:
In this case, the train has a time window of 60 minutes to wait for the bus to arrive and stay for 'x' minutes. Therefore, the bus can meet the train during the time window of x to 60 minutes past the hour. If the bus arrives within this time window and stays for 'x' minutes, they will meet.
Now, let's calculate the probability of the bus and train meeting:
The probability of the bus arriving before the train can be calculated as (50 minutes) / (60 minutes) = 5/6.
The probability of the train arriving before the bus can be calculated as (60 - x minutes) / (60 minutes) = (60 - x) / 60.
Since these two scenarios are mutually exclusive (either the bus arrives first or the train arrives first), we can calculate the overall probability of the bus and train meeting as the sum of the probabilities in both scenarios:
0.5 = (5/6) + [(60 - x) / 60]
Now, let's solve this equation to find the value of 'x':
Multiply both sides of the equation by 60 to eliminate the denominator:
30 = 50 + 60 - x
Rearrange the equation:
x = 80 - 30
x = 50
So, in order for the bus and train to meet with a probability of 0.5, the bus needs to stay for exactly 50 minutes.