Thursday, August 10, 2006

Kittens - a start!

Well, I hope one or two of you are enjoying thinking about kittens and probabilities! It's almost time for me to head on vacation, with little time to finish posting (I'll be back in a week and a half). Let me start with the answer to questions 1a and 1b: the probabilities that one kitten is male, the others female, or vice versa. See if that starts any further thinking on the other questions.

One important hint: think binomial distribution. Now it's not as nasty as it looks on the link.

Without all the math, look at it this way: suppose we're just adding random kittens to a litter. A binomial distribution happens when you have a 50% probability of an event (in this case, say, adding a male kitten to a litter) - and that event has the some number of opportunities to happen. The probability of of having n males after I add a new kitten is logically the sum of probabilities that:

  • we had n-1 males in the litter before adding the new kitten, and now I added a male (50%), plus
  • we had n males, and I added a female (also 50%).
  • So basically, you start out with 1 kitten : its 50/50 that it's male, right? With two kittens, you have 25% chance of no males (because you couldn't have n-1 to apply the first bullet item, right? All that's left is the original 50% chance we started a female, times a 50% chance I don't pick a male this time); then 50% chance of one male (apply both bullet items); 25% chance of two (because for two you can't apply the second bullet item). The totals, predictably, add up to 100%

    So for three kittens (our case), you take that distribution to the next step. From the chart below, the chance of only one male is 3 out of 8 - 3/8. The chance of two males: also 3/8. The answer to 1a is 3/8. And the answer to 1b is 3/8 + 3/8, or 3/4.

    There's some cool math behind it when you get into the details, but I'd better stop there!

    See y'all late August, ok?

    Table showing probability of n males in a total population. Divide the numers under the columns by the "total chances".




    population0 1 2 3 4 5 Total chances
    1 1 1 2
    2 1 2 1 4
    3 1 3 3 1 8
    4 1 4 6 4 1 16
    5 1 5 10 10 5 1 32

    Wednesday, August 09, 2006

    Weep for the Children - part II

    I geared up for my ride to work today, feeling a refreshing, almost Autumnal chill as I stepped out of the garage and into the muted light of pre-sunrise dawn – I’d surely need a jacket and gloves this morning. Attaching my helmet, I noticed the grass in the lawn, each green blade – as if bowing to its creator – slightly bent with tiny hoary droplets of dew. And minutes later as I was on my way, I glanced over to see a wooded hillside with a compact, smoke-like ball of fog. Yet its glory brought a new chill to me: in so many ways the rising fog looked like wisps of smoke, and reminded me of the fresh images of bomb drops on northern Israel that I had seen on the news just moments before leaving.

    I realize this is a similar opening to the one I’ve used before; yet for good reason: my feeling of privilege and awe mingled with the urgent sense of needing to cry out for those in turmoil was almost the same as the other week. “Kittens” solution is going to have to wait another day. I want to fast-forward to the bottom line and ask you to pray or weep for the children of the horrible conflict between Jews and Arabs.

    "But it seems hopeless..."

    One of my problems in praying for world-wide events has often been based on two overwhelming facts:

    • God is in control anyway, and

    • I’m so small.



    Perhaps this fatalism is writ even larger when the antagonists are Zion and Ishmael – with animosities recorded back to Biblical times. I’ve wondered, and perhaps you have too: is this merely setting the stage for a final conflict in which all the world is arrayed against Israel? I don’t ever want to be predicting apocalypse – we don’t know any of the times or seasons – and this may be a small piece of what is foretold. But certainly world opinion, and especially that of those who surround Israel, is becoming monotonically negative towards her with every calculated or errant strike made in Lebanon. Mistakes are being made, and capitalized upon; the propaganda machines are in full tilt. Certainly if photographer Adnan Hajj managed to slip some slanted, doctored photos by Reuters, you can be sure this is the tip of the ideological iceberg. We hear daily of Arab sentiment (astonishingly) turning towards Hezbollah – we shake our heads in dismay as the roots of this bitter weed of resentment grow deeper.

    And our local editorial pages are filled with predictable sneering contempt for Israeli politics from those on the more liberal side of our river. Sometimes I wonder – if the U.S. elects one of the top Democratic contenders in ’08, will the U.S. also become increasingly insensitive to Israel’s plight? Would the tenuous support in England also evaporate?

    While I don’t think Israeli leaders did the right thing at the right time, her plight is unmistakable, as is the horror of the hundreds of thousands of innocent Lebanese civilians now caught in a crossfire largely enabled by the ineptitude and corruption of their government - and of the impotence of the U.N. peacekeeping force, which has allowed Hezbollah to incubate under their oversight for over two decades (see comments by Timur Goksel, “who watched Hezbollah grow into a potent force during 25 years as a senior adviser to the U.N. observer force along the Israeli-Lebanese border”)

    Praying God’s will

    The temptation, then - if it’s all foreknown, if it’s all “God’s will” – is to move on with our merry lives, ignore the pain and suffering; “hope” that it’ll all go away, root for one side or another, and quite frankly, if we’re honest with ourselves, become secretly gratified that it’s not us in the conflict. A heart calloused by years of feeding upon devastation will resort to the one sin we can most easily excuse – apathy.

    I suggest that even a world-sized circumstance – like this one, propelled by irascible popular militias and bumbling, error-prone governments; or where it almost seems that the hand of God himself is bent upon destruction – is not more foreknown or foreordained, than the daily issues we can pray so readily about. I suppose I have to agree with Yoda here – “size matters not.” Indeed all things are fully set under God’s sovereignty – yet we are commanded to pray. Why?

    A recent sermon by our senior pastor, Pat Curtis, on John 11 showed Jesus’ compassion. Here we see a Jesus facing Lazarus' sisters, fully planning to raise Lazarus from the dead, but dealing first with their belief and understanding. What strikes us as odd, particularly since as the readers we know of Jesus' Lordship, is that Jesus, in verse 35, would weep. He certainly wouldn’t be weeping for Lazarus, who Jesus knows is about to be brought to life!

    Pat’s suggestion, and the only logical answer I can see, is that Jesus is deeply moved in great compassion for these people – weeping over the devastating effects of sin on human life. In a kind of groaning which knows no words, perhaps we can see Jesus weeping over the unbelief in the crowd; over the searing pain of death and disease, caused by our sin; but even more so, perhaps it’s in part because now that the Light has come to the world, He knows we are even more without excuse for our unbelief and rejection of Him.

    God’s will is that none would perish. The more I can pray in his will, the more I believe I will find myself weeping for those caught in the path of devastation; and the more I’ll be praying for God’s intervention in the lives of as many as who would come to him. The more I can pray in his will, the more I’ll pray that the cycle of hatred will stop with this generation; and the more I’ll know that God’s sovereign plan will leave nobody behind who can be saved.

    It doesn’t “feel” like much, but our human heart is a liar. God says that “The prayer of a righteous person has great power as it is working.” (James 5:16). That refers to us in Christ Jesus!

    I hope to talk more about the apparent (yet I believe false) dilemma between God’s unchangeable sovereignty and our prayers’ ability to exercise “great power.” In the meantime, I’ll pray for those caught in this cycle of violence, loss, and festering hatred. Please join.

    Friday, August 04, 2006

    Kittens!

    Warning: this post contains probabilities - if that was you I just heard screeching, "ick," please avert your eyes and take this remedial science course.

    Three little kittens! Our nursing foster cat, Sadie, has been so eager to have us handle and enjoy them, so of course the inevitable happened: my wife and daughter started to guess their sex at only three days old (the kittens, that is, were at three... oh, never mind!)


    They think it's two girls; one boy. They keep trying to show me why; I squint, adjust the reading glasses, and after searching for long enough to elicit a frustrated sigh, say something like, "cool!" (Which actually, translated, means, "duh - I don't get it!"). How can they see the machinery in something that tiny and fuzzy???

    I'm sure they're right, but for fun let's assume that my "duh" is well-founded and that the probability of each kitten's sex is completely random - 50/50 - as far as we can tell right now. What are the chances my daughter will be correct after all? Easy one, right? Each kitten has a 50% probability of being the sex assumed; the likelihood of success for each one is an independent variable; so success is 1/2 * 1/2 * 1/2 = 1/8.

    Question 1:
    But we'll open it up a bit (now we get to class 3 of stats 101): what is the probability that, of the three, two will be girls and one a boy - it doesn't matter which kitten is the onion in the petunia patch. (1b:) For that matter, what is the probability that we'll have two of one sex, and one of the other?

    That's still awfully easy for the stats student, but here's where it gets fun: suppose my daughter guessed that we'd have two of one sex, one of the other; while I insisted they'd be either all boys or all girls. It gets ugly; we make a bet. On the day of reckoning, with friends and family and a local film crew (hey, it has to be big, right?), she pulls out a kitten and shows the world which kind it is. I, being on film now, don't want to give up my bet one moment sooner than necessary. So even if I'm wrong, I look in the litter, and if possible pull out another kitten of the same sex.

    Question 2:
    Now, assuming that all that just happened, what is the probability that the final kitten is of the same sex?

    Was that easy or hard? If you're really on a roll, try it the other way.

    Question 3: the bet is reversed with me saying there'll be two of one sex, one of another; I'm still a competitive jerk (just for illustration purposes, mind you - hey, stop that laughing!). So when I reach in to find a kitten I'll try to pull out one of the opposite sex from what she pulled, if possible. If I'm able to do that, the bet's resolved, but what are the chances that the third and final kitten matches my daughter's kitten's sex?

    So, I'd love to hear guesses and thoughts!

    Hint: A portion of this scenario is actually an isomorph of another scenario that rocked the statistics community for quite a while (Question 4: what scenario was that?)

    If there is any interest (and the correct answer doesn't appear), I'll post the answers next week.

    Thursday, August 03, 2006

    Another stop on the way...

    Interesting thing about blogging: while on one hand the fact that others may read makes it a slower process than just slammin’ ideas onto a notebook; that care in delivery makes it palpably more enjoyable!

    It all started with a few thoughts about praying for Israel/Lebanon that I figured I’d write in a couple days; but in order to do that I realized I should back up a bit to my beliefs about prayer itself – another few days. Recently my friend, heavy Dluxe, brought out several very good comments regarding Calvinism (call it determinism if you wish) and our going to God as a father. It actually is a great segue to where I’d like to go – ultimately intending to show that there is in reality no conflict between determinism, free will, and our privilege to call upon God for all our needs. But it means a little more prep-work!

    Oops - now you know where I’m heading! So let me give you a jump start on where I’ve been. As I looked for links to help describe what I believe about the science and theology of determinism and free will, I found this, by Norman Swartz, and this, by Ted Honderich, which match a lot of my thoughts.

    Maybe I’m delving into things too wonderful for me, so I’ll study a bit and take it slowly. I hope to discuss determinism and free will, followed by the amazing power of prayer, and finally (assuming there’s still a crisis to talk about by then), the implications for our prayer life in that area.

    Please continue to pray for the conflict with all your might.

    Meanwhile I have a few other thoughts of a lighter nature (with some fun science and math) that I don't want to keep bottled in while I research this – in particular, my next post will discuss - kittens!

    In Christ,
    Alan