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Subway trains on a certain line run every half hour between midnight and six in the morning. Find the probability that a person entering the station at a random time during this period will have to wait at least twenty minutes. 
  • a)
    1/2
  • b)
    2/3
  • c)
    1/3
  • d)
    1/6
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
Subway trains on a certain line run every half hour between midnight a...
Waiting time of person will lie in the range (0,30) and in this range favourable portion is (20, 30), so the required probability is the ratio of the time interv l's length i.e. 
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Subway trains on a certain line run every half hour between midnight a...
Waiting time of person will lie in the range (0,30) and in this range favourable portion is (20, 30), so the required probability is the ratio of the time interv l's length i.e. 
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Subway trains on a certain line run every half hour between midnight a...
To find the probability that a person entering the station at a random time during the period from midnight to six in the morning will have to wait at least twenty minutes, we need to consider the frequency of the trains and the possible waiting times.

Let's first determine the frequency of the trains. The trains run every half hour, so there are two trains per hour. The period from midnight to six in the morning is six hours, so there are a total of 6 * 2 = 12 trains during this period.

Next, we need to analyze the possible waiting times. Since the trains run every half hour, the waiting times can be either 0 minutes (if the person arrives just as the train is about to leave) or any multiple of 30 minutes (if the person arrives just after a train has left).

Now, let's calculate the probability that a person will have to wait at least twenty minutes.

- The probability of arriving just as the train is about to leave (0 minutes waiting time) is 1/12. This is because there are 12 trains during the period, and the person can arrive at any time when a train is about to leave.
- The probability of arriving just after a train has left (waiting time of 30 minutes or more) is 11/12. This is because there are 12 trains during the period, and the person can arrive at any time when a train has just left, except for the last train.

Therefore, the probability that a person will have to wait at least twenty minutes is equal to the probability of arriving just after a train has left, which is 11/12.

Thus, the correct answer is option 'C' (1/3).
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