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Subject: Probability question rss

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Matt Price
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Over the course of a calendar year, I'm trying to detect a block of 10 contiguous days, say a random 10-day sale at coolstuffinc. I'm allowed to check four times over the year.

What's my % chance of picking a day that falls within that sale period? How does this change if I'm allowed to check monthly (12 times)? how does this change if the sale period shortens to 5 days?

This seems like it shouldn't be a tough question, but it's Friday and I guess I haven't had enough coffee today. Can anyone help?
 
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Andrés Santiago Pérez-Bergquist
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You need to clarify your problem definition. I think what you're asking is as follows: You've got a span of 365 days. Somewhere in those 365 days is a span of ten continuous days, chosen uniformly. (Uniformly over all spans contained within the 365 days, meaning that it starts at latest on the 356th day, or uniformly over all start dates, meaning it can extend into the future year?) Then you want a strategy to pick four days ahead of time when you can check, and want to maximize your probability of one of those days falling within the ten-day span?
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Jim Hansen
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I think it would just be NxL/365 where N is the number of checks per year and L is the length of sale.
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Matt Price
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Thanks for the follow up, good questions.

The 10-day span must have at least one day in the calendar year, but may begin on Dec 23 of the year prior, or as late as Dec 31 of the year. The start day of the span is random.

Rather than a strategy to maximize my probability to select a day that falls within the 10 day span, I'd like to know what's the % chance of me picking the right day to check up on the sale. Say I check at the end of each quarter. Or on the middle day of each month. What's the maths behind trying to get at these %ages?

Thanks!
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Andrés Santiago Pérez-Bergquist
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Any set of check-dates that are at least 10 days apart will be equally good at achieving the maximum chance of success. (Each check will detect any span starting up to 9 days ago up through starting on that day. If the checks are at least 10 days apart, the set of possible spans they can detect is disjoint, so there's no wasted checks.)
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Walt
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Presuming the sales don't overlap, you have 40 winning days and 325 losing days. The chance a single check will hit a sale is 40/365≈11%. Assuming all dates as equal (which is probably false), it doesn't matter when you check.

When you get into multiple checks, thing get more complicated. If you check on 6/10 then you'll hit on sales starting 6/1 to 6/10. If you then check on 6/11, obviously you'll only hit a sale that starts 6/11, the earlier dates being eliminated by your previous check. So, you want your checks to be at least 10 days apart.

The chance of checking four times, properly spaced and not getting a hit is (1-11%)⁴≈63% (the chance of not hitting each time, multiplied together to get the chance of not hitting all times), so your chance of a hit is 1-63%≈37%.

But because the sales are probably not random, this analysis is flawed. For example, if we assume they have a ten day sale at least once a quarter (91 days), then checking at days 10, 20, 30, and 40 in the quarter leaves only days 41-91 uncovered for about a 43% chance, but you'll never hit two sales in a year as you might in the previous case.

More practically,
* Subscribe to the Hot Deals forum
* Use a browser add-on to pop up the CSI page (or remind you to check it) every 10 days
* Maybe, subscribe to their newsletter.
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Tall_Walt wrote:
But because the sales are probably not random, this analysis is flawed. For example, if we assume they have a ten day sale at least once a quarter (91 days), then checking at days 10, 20, 30, and 40 in the quarter leaves only days 41-91 uncovered for about a 43% chance, but you'll never hit two sales in a year as you might in the previous case.

Most sales definitely do not happen randomly. Probability significantly increases if you check during the major US holidays with the highest probability going to the Christmas/End-of-year season.

Tall_Walt wrote:
More practically,
...
* Maybe, subscribe to their newsletter.

Yes, this. You'll get updates to any sales and you KNOW when to go to the web site.
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Tucker Taylor
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mattprice wrote:
The 10-day span must have at least one day in the calendar year, but may begin on Dec 23 of the year prior, or as late as Dec 31 of the year. The start day of the span is random.

Rather than a strategy to maximize my probability to select a day that falls within the 10 day span, I'd like to know what's the % chance of me picking the right day to check up on the sale. Say I check at the end of each quarter. Or on the middle day of each month. What's the maths behind trying to get at these %ages?

(Assume it's not a leap year.)

In a given year, there are 374 possible time periods for the sale. Any given day in the year will be included in ten of those. So, for any given day, the chance of being in the sale is 10/374, or about 2.67%.

If you pick multiple days, it gets complicated.

What you're really trying to find is the probability of picking, say, four days (one a quarter) that *don't* fall during the sale, and then subtracting that from 1. The chance of one day not being in the sale is 364/374. At that point you've eliminated ten of the possible sale periods...

aaaah and then the math eats my brain and I have no idea anymore, because I start working with probabilities greater than one and that can't possibly be right. ANYWAY. That's where you start from. I am certain of that much.
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TVal Thunderstriker

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If you stipulated that the sale date happened completely within the calendar year, the odds of a single pick would be:
10/365, or approx. 2.74%
Four picks (including duplicate selections) would be:
10/365*4, or 10.96%

Since you stated the sale dates only need to include at least one day within the calendar year, the odds change to:
(3650/374)/365, or approx. 2.67%

NOTE: 3650 = 2*1+2*2+2*3...2*9...(374-18)*10, using 9 extra start dates

or four picks (with dupes) of:
(3650/374)/365*4, or [b]10.70%
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TVal Thunderstriker

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Well...somewhere the rest of my summary I typed out got eaten during my previews before submitting and I am not going thru the effort to retype it all from scratch....

Sorry....too much effort for a Friday...
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Matt Price
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You guys are all awesome, this is very helpful.

As an aside, I used the "sale" example (and yes, it would be most advisable to simply subscribe to get updates!) but the true example was for a cohort study with the aim of identifying acute HIV infection. When you first get HIV infection there's a latent phase, where infection is not detectable. This typically lasts about 10 days. Then there's a phase (of about 10 days also) where HIV RNA is detectable but antibodies are not - typically called acute HIV infection. During this phase, you've very infectious, sometimes sick, but "HIV negative" by conventional (antibody) assay.

I was curious to get a conservative/low estimate of how likely we'd be to identify acute HIV infection if we followed a cohort of at risk persons quarterly (or monthly). I should know this, but was having a bad brain fart today (early conference calls can be crushing...)

We counsel our volunteers to come in for RNA testing if they think they've had an exposure (all folks are counseled to be safe, use condoms, etc. but we still see some HIV transmission) so our experience has been to detect acute HIV infection slightly more often than this model predicts.

Again, thanks to everyone who's pitched in to offer comments.
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Walt
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What kind of test do you run? If it's blood, you probably can't get subjects to test every ten days, unless you just need a drop, like a diabetic does. If it's spit into a vial weekly and drop them off monthly, then you can probable get close to perfect coverage.
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Matt Price
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It's a lateral flow immunochromatographic test. A bit like pregnancy tests - just put a drop of blood, in this case, on the tip, add some buffer, and wait ten minutes or so as the capillary action drags it up a strip. If there's a stripe that appears, the person is HIV antibody positive. However, the antibodies appear in response to the steep spike in HIV viral load that typically comes on very early in infection - during this time many folks have flu-like symptoms, but they're also very infectious (so it's a dangerous time to be "antibody negative" and appear uninfected)

There's tests to detect the virus (rather than the antibodies one produces to combat infection), but they tend to be expensive, hard to do, and time consuming. This year and next, we're hoping this will change, as a number of faster, cheaper tests are coming on the market (I think there's already one, maybe) to detect virus.

Some of our peers have done studies that test volunteers *twice weekly*. Very ambitious studies (and expensive), the one we collaborate on provides life skills and job training for women in South Africa twice weekly for a year. Their class schedule starts with a counseling and testing session everyday. Really cool stuff, but would be rather hard to scale up to a national or even regional program!
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