Friday, July 22, 2011

Heads or Tails

I usually give Polly and Pip a bath three times a week – on Monday, Wednesday, and Friday mornings. While they do most of the bath stuff together, there are a couple of differential moments when some kind of choice between the two of them must be made. The first moment comes at the beginning when we have to decide whether or not to use bubble bath. The second moment arrives in the middle when I have to identify who will get their scrub-down first. The third moment arises at the end when one child must get out and be dried and clothed while the other is allowed to stay in the water and play a bit longer.

For Monday and Wednesday, these differential moments are easily handled through the taking of turns: on Monday its Pip’s turn to select the type of bath they will have, to get washed down first, and to stay in the water longer; on Wednesday its Polly’s turn to do these things. Friday, however, presents a dilemma. On Fridays I get to choose whether or not to use bubble bath (usually not, since it’s almost impossible to wash the bubble bath suds out of the kids hair), but I don’t want to have to keep track from week to week which child got to go first and stay in the bath longer. We already have too many of instances of turn taking that I have to keep straight as it is and with the time interval being relatively long, I just wind up getting confused about who did what when. So, I decided to flip a coin instead.

Now, the coin flip is a decision-making technique with which I have an ambivalent history. My parents used it occasionally to resolve competing claims between my sister and me over who got to sit in the preferred seat in the car or who got to choose what music we would listen to. I remember the coin flip being a constantly frustrating experience for me because when it was her turn to choose heads or tails, my sister would always take the latter and win. When it was my turn, I would guess one or the other and generally lose. This sense of being beaten down by the gods of chance was only sharpened by my inability to complain or appeal to anyone. Of course, it was these very qualities that made the coin flip so appealing to my parents and why I was happy to inflict this exercise upon Polly and Pip.


To make the whole coin flip a bit more of a production, I developed a ritual that turns the thing into a lesson in probability. To start with I tell Pip and Polly that there are two sides to the coin, heads and tails. Then I show them what each side looks like. Next I tell them that because the coin is evenly weighted, there is an equal chance that after being tossed in the air, the coin will come up heads or tails. Then I add that, as we do this week after week, the coin will come up heads and tails approximately the same number of times, meaning that over time you each will get to stay in the bath longer about the same number of times. Finally, I ask one of them to call it in the air.

Now those of you who have some experience with probability might notice a problem with this ritual. While my description of the probability at work in the single coin flip was correct, my characterization of the long-term results was not. In order to get the long-term evenness between heads and tails that I was describing to Pip and Polly, the only thing that can be allowed to vary is the flipping of the coin. But, by letting Polly or Pip call heads or tails, I introduced a second variable. This second variable means that in any given flip there are four possible results – child selects heads, coin lands heads; child selects heads, coin lands tails; child selects tails, coin lands heads; child selects tails, coin lands tails. While within these possible results there is still an even chance between ‘wins’ and ‘losses’ for a given flip, the second variable – the child’s choice – does not possess the same evenness in probability as the flipping coin. In fact the child’s choice must be considered completely random in that there is no way to predict over a series of flips how many times the child will choose a given side of the coin. This means that the win/loss balance for this series of flips will also be completely random. The fairness that I promised to Polly and Pip was a lie.


It took me about four weeks to realize my mistake. At that point I made the easy fix and permanently assigned heads to Pip and tails to Polly. These will be their assigned sides from now until I no longer have to arbitrate these choices for them.


I don’t know that Polly or Pip will ultimately appreciate the amount of consideration I have given to this otherwise insignificant moment in their Friday morning routine, but it feels like a small victory to me. In my daily work with Polly and Pip I don’t often get to put aspects of my formal education to work in such recognizable ways. There was something satisfying in doing so, in taking a stab at something, sensing that there was a problem with my approach to it, and then working out from my memory what I needed to do to fix it. It was my own little internal game, one of which Polly and Pip will never be the wiser, and it made me happy that I got it right.


Christopher said...

I don't see the problem here. Conditional on the child choosing heads (or tails), the probability of winning is 50 percent. Since the child's probability of winning is the same either way, why does it matter if you select their side or if they choose themselves? Regardless, they win half the time, which seems perfectly fair.

Jason said...

I think there IS a very subtle difference from a psychological standpoint. As Christopher above mentioned, mathematically they should still get the choices evenly over time. However, psychologically it "feels" different if in the first scenario your sister got the reward because you failed to choose the correct side where, in the second scenario, it seems the "gods of chance" chose your sister over you.

In a very strict sense there is no difference between the two scenarios; in both cases you lost because you didn't "choose" the correct side. The difference is that in the second scenario your choice has been locked in; the outcome is solely based on the coin toss. Our brains do not understand probability very well (at least not intuitively), so these seem like two different scenarios (which is also why you hated the activity when you were a child - it seemed your sister was getting rewarded because of your inability to choose correctly, when in actuality your choice didn't matter in the long run).

It might be more useful to start out with the second scenario, as you're doing now, and then when they get older and get better at thinking about probability, to switch back to the first scenario and see if they can reason out why there's no inherent difference between the two methods of choosing.

Anonymous said...

I never flipped coins very much to choose such things with my two. It never seemed to appeal to them and never registered as a "fair" way of doing things. Often sticking to simple turn-taking worked best.

However, as they got older (by 4-5, certainly) and were able to do the problem solving, we let them sort out issues like this on their own more and more. We might set some rules, like "it has to be something that you both can live with", but left a lot of it to them.

The results were sometimes surprising. Rock-paper-scissors became a favorite way of settling disputes - I think the fact that it was a game was part of the appeal. In fact there were times when my daughter (the older) was not particularly invested in the decision and would conced to her brother, only to have him insist that they settle it with rock-paper-scissors to make sure it was fair.

Now, at 9 and 11, they settle nearly all disputes about "whose turn" or "which movie" or the like on their own, usually with a lot of generosity, and find adult intervention to an intrusion more than a help (though adult intervention is always there as a fallback if the dealings aren't going fairly).

But it sound like whatever the statistics involved, you have a system that works and feels fair to both kids - which is ultimately more important than the purity of the math.


Rick Juliusson said...

Christopher's right that statistically you still have equality, and Jason's even more right that the important thing is that the children BELIEVE it's fair.

My boys are now 7 and 9 and too big to share a bath, but they do share bath water, so the question of who goes first is a big one. I still make the mistake of letting them remember whose turn it is, or succumbing to the younger one who will throw a 20 minute tantrum and make us late for school. Time to break out a coin - thanks!

Anonymous said...

I wonder if it's easier to just write their names on the coin. No heads or tails, just names. Thanks for the idea. I will definitely it. Of course, this could get messy if there are 3 kids.