July 15, 2009

PLEASE, MONTY HALL, LET ME SWITCH FROM GOAT TO BABY



Last week, I got obsessed with probability problems.  Since my likelihood of having a successful pregnancy is 50/50, it mirrors coin toss statistics.  I originally wrote that I had a 6.25% chance of a fourth miscarriage, but I kept thinking about the problem and realized that was too simplistic.  That assumed that I had four chances to get pregnant and struck out on all four.  But that wasn't the case: in fact, I had 22 chances to get pregnant, got pregnant on four of them, and flipped tails on the first three.

One of my good friends is a statistician, so I contacted her and asked her a coin toss problem:  Let's say you flip a coin 22 times and mark down whether each flip is heads or tails.  What is the probability that I could choose four random flips -- say numbers 8, 13, 19, 22 (the months I got pregnant) -- and have them all be tails?

She thought about it and replied:

I think the answer to this question is:
 
(11 choose 4) / (22 choose 4) = 330 / 7315 = 4.511%
 
Explanation: The number of ways to select 4 tails out of 11 possible tails divided by the number of ways to select 4 coins out of 22 possible coins.
 
(11 choose 4) = 11! / (4! * 7!) = 330
(22 choose 4) = 22! / (4! * 18!) = 7315

My gut feeling was right that the probability was even lower than it would've been with just four coin tosses (though I know enough about stats to know my gut isn't always right; otherwise, we'd all get goats.)

Anyway, I've been obsessing about this for a while.  I would start to feel confident that surely I wouldn't get so unlucky again, but then I would reign myself back in.  When it happens to someone theoretical, it's a statistic; when it happens to you, it's a tragedy.

In the car on the way there today, I was certain I would get bad news.  I was a wreck.  I had to share another ultrasound room, which nearly sent me into a rage until I realized the other girl was only there to check her follicles.  Luckily this time, it seemed that the hospital staff actually knew who I was and knew to tread lightly.  They were all nice and at least acted like they had read my chart five seconds before coming in the room.  They were sorry my husband was gone and expressed hope that this time would work out better than the others.

And it appears that, for now, Schroedinger's cat is alive.

Luckily, their sympathy extended to extra medical attention too: I get to have weekly ultrasounds.  I go back next Wednesday morning to see if the baby is still alive then too.  If it's still alive in two weeks, that will be the furthest I've ever progressed.  I won't begin to feel relaxed at all until then, but for now, I'll take whatever good news I can get.

And I can go back to feeling bad and sad.

Posted by: Sarah at 10:48 AM | Comments (20) | Add Comment
Post contains 527 words, total size 3 kb.

1

You know I'm around no matter what.  And, God love ya, you knew better than to ask this girl about math.  Watched me struggle one too many times to figure out a decent tip, eh?

Lurve you!

Posted by: Guard Wife at July 15, 2009 11:29 AM (M+hWl)

2 So excited for good news! And that I thought to check your site!

Posted by: Wifeunit at July 15, 2009 12:11 PM (+2Ggg)

3 You were right the first time. That is, the chance of getting four tails in a row is 1/16, or 6.25%. The other flips, that you don't look at, don't matter, just like the ones that have already passed don't matter. The chance of getting tails next time is 50%, just like every other time, no matter what you flipped in the past. 

Posted by: David Boxenhorn at July 15, 2009 12:26 PM (AZIQI)

4 All those deep statistical thoughts are giving me bad college flashbacks!  I am relieved for you (at least until next week).  Thank goodness the medical people finally have their act together over there.  

Posted by: dutchgirl at July 15, 2009 12:45 PM (hLAkQ)

5 Yes, David, but it's not four flips in a row. Because you have to consider the fact that I put out one egg every month, but I don't get pregnant every month.

There are two things at work here: the probability of GETTING pregnant and the probability of STAYING pregnant.

Yes, my odds of getting pregnant in any individual month remain the same as any other human female: 1 in 4.  I have roughly borne that out with my 4 pregnancies in 22 ovulations.

Once I am pregnant, the odds of staying that way are the 50/50.

I'm considering "eggs" as flips, not "pregnancies."  Each egg is a coin toss.  Each pregnancy is more complicated than that.  And the flips that you don't look at DO matter if the question you're asking is "what are the odds I still haven't had a healthy baby after 22 ovulations?" instead of "what are the odds I'll keep this baby I am carrying right now?"

I know each successive egg has no bearing on the next, just like coins don't, but cumulatively, I should've theoretically put out 11 good eggs over these past 2 years, but none of those managed to fertilize. So it's more than just four flips, cumulatively. It's 22 flips.

As my stats friend said, picture this:

Let's say you have these two different sequences of 22 coin tosses (keeping with your example of T on the 8th, 13th, 19th, and 22nd tosses):

1) H T H T H T H T H T H T T H H T H T T H H T

2) H H H H H H H H T H H H T H H H H H T H H T

The probability of randomly selecting 4 tails out of these two sequences is different.

Even though both scenarios could be what has happened to me over the past 2 years, I should theoretically be closer to the first.  (Setting aside the fact that we have no way of knowing if my eggs are evenly distributed at all, that is to say that 25% are normal, 25% are balanced, 25% are unbalanced 7, and 25% are unbalanced 22.  It should work out in theory over 100,000 eggs, but who's to say that it really does?)

I am no math expert, but I respectfully think you're wrong.  I think it's more complicated than just four chances...

Posted by: Sarah at July 15, 2009 01:03 PM (TWet1)

6 I read this post very slowly, fearing what was at the end. The graphic at the beginning asks, no, shouts the question, So which is it?

I saw the problem the way David did, assuming that past outcomes had no effect on future ones.  However, at the moment, I think it's more complicated than that. What's the chance of you being pregnant in any given month? I'm no statistician, but I doubt it's just 50% - or that it's even calculable. The probability of your current situation involves that figure, whatever it is, as well as the 50/50 per pregnancy.

Thank you for reprinting the details of how your friend got the answer instead of simply giving us the answer. I may use (x choose y) methodology for my own coin toss problems in the future.

I am impressed by your mental versatility which is evident even from the last few posts. Politics, poetry, statistics ... what don't you do?

Posted by: Amritas at July 15, 2009 01:07 PM (+nV09)

7 The math threw me, bring the statistician to the gulch..just saying. 

Prayers continue, and all of that. 

I keep thinking and praying for each of you....

I know you want to move forward, and I know you want future, and i know you want to start walking forward..


Posted by: awtm at July 15, 2009 02:28 PM (mYtP/)

8 Sorry, Sarah, I misunderstood your question. I thought it was "what is the chance of getting tails four times in a row [assuming that you flip the coin four times]".

I don't understand what your question actually is, though. Now it seems to me like it's "what is the chance of getting tails four times in a row when I try to flip 22 times, and the chance of successfully flipping is 1/4"? But that question really doesn't make sense to me because it seems like the interesting question would be "what is the chance of getting all tails when I try to flip 22 times, and the chance of successfully flipping is 1/4".

Posted by: David Boxenhorn at July 15, 2009 02:31 PM (AZIQI)

9 First, *whew*. I am glad today's news was not bad. I won't say 'good' simply because it has not allowed you to relax. I will continue to pray and look forward to next Wednesday. I'm quite familiar with waiting/living in limbo and I know how exhausting and frustrating it can be. My prayers are for peace and strength (in addition to an incredibly boring and textbook-perfect pregnancy) for you and dh.

Second, you are right...there are two variables at work in this situation. Getting pregnant and staying pregnant. And, while your previous pregnancies have no true bearing (in a statistical sense) on your current pregnancy (or any subsequent pregnancies), they do establish a (really crappy) pattern.

For simplicity's sake, I'd round to the middle and say 5%. Which is better than 0% (for the 'glass is half full' population here) but truly sucks monkey balls (for those with a more practical outlook on life...some might call us cynical) when all you really want is a baby.

Sorry to ramble - I've been working in my Calculus study guide and my brain is stuck on numbers. I'm praying hard and looking forward to hopeful news next Wednesday. (I may be in a cynical mood these days but I'm still an optimist at heart).

Posted by: HomefrontSix at July 15, 2009 02:32 PM (7Qxzl)

10 David -- Now I am the confused one!  I doubt that I got 22 tails, meaning 22 bad eggs in a row.  I am guessing that somewhere interspersed in there were good eggs.  I'm making this up, but like potentially in January 2009, I had a good egg, but I just didn't happen to get pregnant that month.  The next month in Feb, I got pregnant, but that was a bad egg month. 

I am assuming that I have had somewhere around 11 good eggs over the past 22 months, and just lamenting the fact that out of those 11 good ones, I never managed to get pregnant during those months.  I always got pregnant during bad egg months.  The probability of *that* seems to be 4.5%, and it gnawed at me for a week.

Anyway, you're right in the sense that none of this matters anymore and I'd just like to concentrate on the 50% chance I currently have!

Posted by: Sarah at July 15, 2009 02:42 PM (TWet1)

11 My eyes glaze over at statistics because when it comes to flesh and blood sometimes they just don't matter. And because I just don't often pay attention to them. One of my earliest favorite books was, and still remains, "How to Lie with Statistics."  That has nothing to do with your statistics but to my attitude to most of them. If I can't do the figures I just leave it to someone I trust, as you did. I guess when it comes to this type thing I take the most optimistic part I can find and go with that one. That's right, I am a woman and we think with emotions first. Not that we can't logic, but sometimes we just choose not to. Easier that way.  I'm waiting for the next week and the week after that, because I am choosing to believe there will be weeks after that. Like I said, it is your turn.

Posted by: Ruth H at July 15, 2009 02:51 PM (4u82p)

12 The question that I stated above as "interesting" translates into, "what is the chance not of getting pregnant with a good egg after 22 tries". The answer I get is 2.2%. Here's my reasoning:

The expected number of coin flips is 5.5 (22/4). The chance of getting all tails is 1/(2 ^ 5.5) = 2.2%. But I think it's wrong since I don't think the chance of getting pregnant is really 1/4.

In any case, you're right: You're past all that now.

Posted by: David Boxenhorn at July 15, 2009 02:52 PM (AZIQI)

13 I'm so happy you got good news. I will continure to pray that the news stays good. I don't know how you do what you do, you're so strong.

Posted by: Mare at July 15, 2009 02:54 PM (HUa8I)

14 David -- You're right: the 1 in 4 number only comes from looking at what happened after the fact.  I think it comes from a lot of data of many different people, and then the stats are kajiggered.  I don't think it necessarily means that every individual woman has a 25% chance each month, because some women are really fertile and others less so...or the same woman with two different men might bring a different result.  I think it's a ballpark figure that works across the board, but it's hard to apply that to any given specified month and to say that in Aug 2009, Betty Sue has a 25% chance of getting pregnant.  Instead, they say that if you try to have a baby for one year, 80% of women will be successful.  But there's a lot of variety in that sample.

Posted by: Sarah at July 15, 2009 03:15 PM (TWet1)

15

Wait a minute, your prego now, 100%. Yea, yea, I know about the last times, forget that for now (with all due respect), your 100% prego NOW. Screw all that and that mind numbing math, the glass is half full...NOW.

Remember, only Liberals and losers (excuse the redundancy) are pessimistic.

Posted by: tim at July 15, 2009 03:28 PM (nno0f)

16

1)  I love the way your mind works...  with the statistics. 

2)  I'm praying my butt off for you and S's cat. 

Posted by: Val at July 15, 2009 05:42 PM (5btL/)

17 Against all odds (and all goofy statistics), life will find a way.  I'm still praying for you.  Was holding my breath through that whole post, and am so relieved to not see bad news.  So here I sit cheering for Schroedinger's cat. . . .  yeah

Posted by: rc at July 16, 2009 12:07 AM (aJX4Q)

18 *cautiously cheering*

I've been thinking of you almost constantly the last few days.  I suspect if all the hope, love and good thoughts I'm sending your way could truly affect fertility, you'd be having octuplets about now...

*holding my breath for you*

Posted by: FbL at July 16, 2009 12:42 AM (HwqvF)

19 Probability is one thing, but how about actual runs of numbers? Since I have access to some considerable computer power just sitting around (and my minions can do my actual work for me as I play with these numbers) I did a quick 10 simulations with the criteria you hypothesized. I did 100 sets of flips, with 22 series of "flips" (zeros and ones) and then randomly picked 4 flips out of each set, marked how many were 4 tails in a row, and did this 10 times. The series is as follows: 6, 6, 9, 6, 6, 8, 7, 12, 5, 6 And all of those numbers mean nothing, because that's all they are, numbers. They don't show you any probability about your current "flip". My flips are cold and calculating and strongly dependent on the RAND function of Excel. Your personal flip has the support of a while bunch of people both known and unknown to you, the pinnacle of human medical knowledge and technology, and of course, you and your husband. You can't calculate how those affect your "flip". In other words, we're pulling for you, despite the odds.

Posted by: deskmerc at July 16, 2009 03:00 AM (pYOXQ)

20

I am too dumb to get the statistics.  But I can relate, definitely, to waiting for the other shoe to drop, as so many have after multiple miscarriages....  You're on my mind!

Posted by: Allicadem at July 20, 2009 07:21 PM (nlT4X)

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