I love high quality discount stores like Marshalls, TJ Max or Filene's Basement. Nothing could beat the joy of finding a treasure for $20-30 without the guilt of overspending. I venture to Marshalls approximately once in two weeks, at a random day and time, spend there around 30 minutes and go home satisfied by my "shopping drug dose".

Slowly I started to realize that around half of the times that I shop there I run into the same mom from my kids' school. We both get slightly embarrassed, assuming that the other must be living there for our encounters to be so frequent. This mom doesn't work there or hang out there, in fact she seems to spend around 30 mins at the store, like I do.

I am wondering whether our high frequency of encounter means that she indeed comes there every day or that her shopping frequency is relatively rare like mine and it just so happens that our in-between-work-and-school shopping cycles and time tables frequently coincide.

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Answers accepted all day long on Friday April 1st, on our Family Puzzle Marathon. They will be hidden until Saturday morning (EST) and everyone who contributed something reasonable will get a puzzle point. Please, explain your answer.
## 9 comments:

There's no arithmetic to do here unless we get hypothetical, which we could, but I don't want to. The times that Maria (or anyone) chooses to go shopping will be much influenced by the other things in real life such as school schedules or bus schedules, weekdays or weekends, sales being advertised, meals with friends, computer habits, circadian rhythms, TV habits, mall hours, and perhaps work.

I get to the gym 3 times a week, and generally encounter many -- at least some -- of the same people, and this is no surprise at all. Our attendance is not really random.

It's not surprising if indeed both shoppers go there "frequently" such as several times a week. The more often they go shopping, the oftener they will coincide. Just a "density analysis" issue, if that's the correct term. Ah, Maria says maybe once in two weeks, for 30 minutes. I wonder if that self-reporting is accurate.

And once the coincidence is noticed by either or both shoppers, then it does become (or seem) more noticeable.

Perhaps both ladies go shopping after the gym, or after manicures?

Hmm, as usual I’ve thought of some more issues, sorry. Maria lives in or near Boston, quite a large city with many such retailers, and that high number would tend to reduce the odds of coincident shopping encounters.

However both ladies have children in the same school, and they probably live in the same zipcode; that will tend to negate the hugeness of Boston, and they both have their favorite stores, who doesn’t.

And, um, both ladies’ shopping days might indeed be influenced by other biological cycles, just sayin’, (“approximately once in two weeks,” she’d say).

Finally (I hope), Maria hasn’t quite said whether the other lady is a shopaholic, perhaps she's really shopping most days. There are such women. Ask her, Maria. Go for coffee.

I think the math would be something like this. Most Marshalls open at 0930, but with school children, some cleanup and drive time, Maria and the other mother are probably not shopping before 10am, ever.

There are 2 good hours before lunchtime (10-12), and then perhaps 2 more good hours after lunch (1-3pm?), before it’s time to head home for the kids’ bus and dinner prep. Gosh, I hope this is not sexist or whatever. But I think their available shop time is almost always limited to 4 hours on any day. Not evenings. And can we assume it is weekdays only, M-F? 20 hours per week. Less time available than one might first think.

Let’s take One Day, first. Today, Friday, and assume both ladies will go shopping at The Same Marshalls Store, for 30 minutes, this morning in the 2-hour span 10-12am. Chances they will be in the store at the same time are….whatever they are, reminds me of an earlier overlap problem I missed, maybe 50%, probably less? Chances they will SEE each other are hard to guess, but even less likely.

One of them could shop this afternoon instead, cutting the odds of an intercept today by half. Or Maria could wait until Monday instead, or Tuesday; if she is really shopping on random weekdays and times, and only twice a month, the odds of an intercept drop greatly. Not probable they would meet so often; it is improbable.

And there are so many stores (and so little time). Granted they live in the same neighborhood, and have similar tastes in style and price. Still improbable.

We’ve got no data about the other lady’s shopping schedule, and indeed that is Maria’s question. I think two things. (1) One or both of them are shopping much more frequently than twice a month, and/or (2) there are other forces/cycles at work such as grocery shopping, weather, coffee, gym schedules, paychecks, advertising, or other BioRhythms (ever read that old book, now debunked?). The two shoppers could very well be more synchronized than they realize.

But still I suspect reason (1). If the other lady is there (wherever) quite a lot, then the odds of an intercept go way up. If Maria shops a lot also...

Or else, Maria, you are being stalked.

I think that she shops maybe, twice a week every 2 weeks, or once a week every week on (a) random day(s). That should provide the circumstances you are noticing.

I try...

The probability distribution is a binomial: they meet or they do not meet.

P=0.5 which is also the mean of this distribution.

Let say that the visiting activity could be from 10 am to 8pm so 10 hours period.

There is (10*30)^2 possible combinations which is 90000.

The chance for the woman to be in the mall is m/365 and n/365 for her friend.

M*n/365 is the expected number of days for both.

The probability for both of them to visit the mall at the same time and day is:

(0.5/90000)*(mn/365)

M and n are in the numerator, if the woman shopping frequency is rare then the denominator will be bigger if her friend has a high shopping frequency.

This problem reminds me of "chances of a random encounter" but we had a definite period of time which we do not have in this problem.

This happens to me quite often. I think that you have a routine and that others do to. When these routines intersect you will experience a higher than coincidental number of encounters.

I go to Trader Joes on Thursdays after I drop my daughter off at a nature class. It's close to the class and the rest of the week it's a little too far out of the way to bother. I often see a couple of the same people there every week because they are probably on a similar weekly schedule.

The woman you see at Marshalls most likely has free time in her schedule at the same time as you.

I did not take into account the sentence " once in two weeks" but I am certainly wrong anyway...

It seems logical to think that the other woman has a high sHopping frequency or yes, she has a stalker:-).

Very interesting.

Tom's questioning of my estimation of the probability of our encounters is absolutely right. It may be 1/10th or 1/5th but it seems much more frequent because it is memorable. Similar to the probability of a crime being vastly overestimated because we constantly hear about it in details on the local news.

But assume my estimation of 1/2 probability of our encounters is correct. Could this just be explained by this woman visiting the store every day?

I called my dad to consult on the solution to this problem and here is what we came up with.

Please feel free to agree or disagree.

Assume that this woman comes to Marshalls on a daily basis. Could this explain our 1/2 probability of encounters? Turns out no!

Marshalls is open 10 hours x 60 mins = 600 minutes during the day. If I spend 30 mins there, it means that this woman should come in the interval -30 mins, +30 mins before or after I do for us to meet. Say, grossly it is an interval of a length 60 mins. Probability of this is 60/600 = 1/10 And we have 1/2. It means that she either should come to the store 5 times during the day or something else is influencing our encounters.

It may be a BioRythm as Tom suggested and I will use this explanation as my next excuse for shopping. Or, more likely, it is just our schedules. My ventures to the store may seem random to me: just before picking us the kids for school, while my daughter has a misic lesson, on my lunch break etc. However, they are indeed influenced by the weather, school schedule, season changes etc and this all somehow leads to such high probability of our encounters. So, I should stop assuming that she is a shopaholic and we indeed have nothing to be ashamed of when we frequently meet.

This wasn't an easy black-and-white puzzle. Thank you all for daring to write your thoughts.

Hopefully this all made us think about about pleasurable math of regular daily activities.

A puzzle point or Tom, Lynnet, anne-marie, thelittlebird.

could be both, cant it? environment probably does have to do with alot but it might be chance

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