Gosh, I can see I’ll have to be much more careful in my future puzzles.

On your first point:

Andy and Bob are counting whole days. Really they are counting the number of ‘slots’ in which the event could occur. So if Bob turns up at 10am and the event occurs at noon the same day then he would count 1 day. If it helps we can have Bob turn up just after noon so his waiting time will actually be whole days.

This puzzle is based on ‘World of Warcraft’. In that game there are ‘quest givers’ who have five different quests to give out. The quest is chosen at the start of each day, i.e. at midnight, and is available throughout that day. A player can take the quest at any time during the day but can only take one quest each day. I found myself wondering how long it would take me to get all the quests, and that turned out to be more interesting than actually playing the game!

A different puzzle would be to have the event occur at any time and measure the actual time Andy and Bob wait. In this case we could have multiple events in a single day. As you say this is modelled by the poisson distribution.

On your second point:

You are right that a short wait will occur more often. I used the phrase ‘one of’ to get round this.

As you point out the actual distribution of Andy’s wait times (the skew) makes a big difference. Independant events will have a particular distribution. Any other distribution indicates some connection between the events and in this case Bob will get a different answer to Andy.

I have calculated the distribution of independent events and put them in a spreadsheet which you can access here.

I assume we do the experiment over 10,000 days with the event occuring every five days on average.

In the third tab, ‘Calculating the Values’, r is the number of days that Andy waits. Column D is the number of times that Andy will wait for this many days.

In the first tab, ‘Large Example’, I use these values to calculate Andy and Bobs’ expected wait times. For Bob I calculate the probability of him appearing during Andy’s wait and what his wait is likely to be. Note that I am adding up ‘expected wait times’ for Bob, not actual wait times.

At the top I have Andy and Bobs’ wait times which are 4.975 and 4.913 respectively. They are slightly out partly becuase of rounding and partly because I only go up to an Andy wait time of 30 days. To be truly accurate I should go up to infinity.

On the third point:

Yes, I am assuming that Bob turns up much less often than the event. It is important each time Bob turns up is clearly not connected to the last time he turned up.

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“The event can only happen once each day, let’s say it always happens at noon. Bob always turns up at 10 a.m.”

This throws me off a little. So Bob’s appearances and the events are each randomly occurring on 24-hour cycles, but slightly shifted. This is the same as if Bob’s random appearances always occur at noon (i.e., in sync with the random events) but then we always add two hours to his wait time. So that causes a problem right there. Bob’s average wait time is going to be 2 hours longer than Andy’s. What if Bob has a brother who always shows up at 9:30? Then he will observe exactly the same events that Bob observes since those can occur only at 12:00 but his wait time will be 1/2 hour longer than Bob’s every time. I think you have to make the assumption that the time of day is the same for both Bob’s appearances and the events, or alternatively, that the probabilities of Bob appearing or an event occurring are equal at all times throughout the day.

“…Bob is more likely to turn up during one of Andy’s long waits than during one of his shorter ones.”

This seems obvious, but might be a little trickier than it seems. Yes, a longer wait for Andy means a larger window of opportunity for Bob, but if there are more short intervals than long intervals Bob might be more likely to show up during a short period. For example, let’s say that for the first 100 days there are 20 intervals of 5 days each. And say the next 100 days is just one long interval. Bob may be just as likely to show up during several short intervals as he is during the one long interval. But, you say, that is just because I constructed this artificially skewed distribution of events. The thing is, waiting times for independent events are usually said to follow what is known as a Poisson distribution which is inherently skewed. This means that if you pick an arbitrary regular period of time, like months or years, and count the random events in each period, you will observe that more periods will contain an above-average number of events, and fewer have a below-average number of events. In other words, there will be more short intervals than long intervals. So *technically* I think your statement may not be true but your reasoning was basically sound. It’s not that Bob *hits* more long intervals than short intervals. It’s that Bob *misses *more short intervals than he does long intervals. Hope the distinction I’m trying to make is not too confusing.

“On average Bob will wait half as long as Andy for any event that Bob sees…”

Maybe. This is true over the long run if the rate for Bob’s random generator is as long or longer than the rate for the event random generator, which I think is what you intended. If Bob showed up more frequently he would usually show up somewhere in the first half of Andy’s wait, like in my first example from yesterday. You were clear about using the Ernest random number generator for Bob, and I think you meant for that to correspond with the average rate of occurrence for the events also.

You’re right, I should clarify exactly what Andy and Bob are up to.

In that example there is only one event, not the five in the main problem. The event can only happen once each day, lets say it always happens at noon. Bob always turns up at 10am. Andy waits for the full duration of the test regardless of how many times the event happens.

How does Bob decide when to turn up? Okay, let’s go with your idea that he turns up when an Ernest is born in California. Are people born Ernest or are they named later? I was going to go with christenings but that excludes people who aren’t christened. Let’s say Bob checks the births section in the Californian press each morning and visits Andy when he reads the anouncement of a brand new Ernest.

The answer to the puzzle is the paradox is that Bob is more likely to turn up during one of Andy’s long waits than during one of his shorter ones. On average Bob will wait half as long as Andy for any event that Bob sees, but he will miss a lot of events that Andy sees which follow on quickly from the previous event. I have done the maths and this does work out. Unlikely as it seems Andy and Bob will have the same average wait time.

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Let’s change the name in my second example to Ernest. There’s one born about every 11 days in California which is closer to Bob’s average wait time. ]]>

If you stop Bob’s clock when he is not present you can see that from his perspective, the calculation of his average wait time is the same as Andy’s.

There is also a second mistake in your reasoning. For the moment, forget that Bob and Andy may be counting different events, let’s just count Andy’s events. It is not necessarily true that over the long-run, Bob will on average appear mid-way between Andy’s observation of events. It depends on how you define the process that generates Bob’s random appearances. It is not enough to say that Bob appears at random times – you have to be more specific. For example, if Bob decides to reappear every time an atom of Uranium-238 decays somewhere in the universe, he will always reappear almost immediately after he leaves. So his average wait time will be very close to Andy’s. (He will also be very unlikely to miss any of Andy’s events).

If on the other hand, the random process causes Bob to reappear a little less frequently, like every time that a boy is born and named Robert in California (about once every 1.5 days), he will be more likely to appear somewhere near the mid-point between consecutive observances as you assumed, though he might also miss some of Andy’s events.

If the random process is very long like say watching a particular atom of Uranium-238, then he will probably miss all of Andy’s events. ]]>