Bus stop
Every morning, Bob goes to work on a bus #5, which arrives at every 10 minutes of the hour. He himself arrives at the bus stop at a random time. There is another line of bus, #6, stopping at the same bus stop. Bob observes that it is more likely that #6 comes first, followed by #5. So he thinks, "What a pity that I cannot take #6 to work. Obviously, it is running more often than #5."
Is Bob's thinking correct? Can we conclude with confidence that #6 runs more often than #5?
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1 solution
It is perfectly possible that #6 runs in every 10 minutes as well. For example, assume that #5 comes at 00, 10, 20, 30, 40 and 50 minutes after the hour and #6 comes at 09, 19, 29, 39,49 and 59 minutes after the hour. If Bob goes to the bus stop randomly, it will be more likely that he is in the bus stop during the 9 minutes when #5 just passed and #6 arrives than in the 1 minute when #6 just passed and #5 arrives. In average he will have 9 times out of 10 bus #6 arriving first. And yet the two buses run in the same 10 minute intervals.
A more relevant measure is how long does Bob waits before the bus comes. If they both run in 10 minute intervals, the average waiting time will be the same for the two buses, even though #6 may come first more often.