Statistical intuitions

This post is about what might be called the intuitive component of statistical reasoning, or perhaps more accurately statistical impressions.

I ran into what for me seemed like a kind of paradox or conundrum when considering the following probability exercise: Suppose you have two jars, each of which have 50 black and 50 white balls, randomly distributed. What are the odds that you’ll get two black balls if you pull one ball out of each jar without being able to see what color it is first? This is a simple exercise: since the odds of getting a black ball are .5 on each of the attempts, the odds of getting two black balls are 25% (.5 x .5), since the probability of two statistically independent events happening is just the probability of each event multiplied by the probability of the other event. So this is like the probability of getting heads on two consecutive coin flips, assuming a fair coin.

But what if there are 80 black and 20 white balls in the first jar, and 80 white and 20 black balls in the second jar? You are still making two random selections from 100 black and 100 white balls, but the odds of getting two black balls aren’t 25%: they’re 16%. (.8 x .2).

On some level that I can’t really verbalize, this somehow seems counterintuitive. I can work out in a formal way why this answer is correct, but it doesn’t feel correct to me, again in some intuitive way. It feels intuitively to me like the odds of getting two balls of the same color ought to be the same in the two exercises, because after all the total number of black and white balls is the same in the two experiments.

This little exercise drives home to me why a lot of people resist the correct answer to the Monty Hall problem. (Paul Erdos, one of the greatest mathematicians in history, refused to accept the correct answer until he was shown a computer simulation demonstrating it empirically!)

I saw the solution to that problem immediately the first time somebody described it to me: for whatever reason, my mind is wired — if that’s the right metaphor, which it probably isn’t — to “see” the solution in both a formally reasoned and psychologically intuitive way. But while I can understand the correctness of the formal solution to the black and white ball problem above, I can’t “feel” it as intuitively correct, which makes it harder for me to on some level accept the conclusion (which again I’m not questioning (nothing the god of biomechanics wouldn’t let me in Heaven for, anyway).

I’m curious whether this description of formal versus intuitive understanding of statistical and/or other sorts of conclusions resonates for anyone else.

Update: Many commenters have pointed out that taking the experiment out to its logical extreme can be clarifying. For example if there are 100 black balls in one jar and 100 white balls in the other jar, the probability of drawing two balls of the same color is zero. This is also a helpful way of “seeing” the answer to the Monty Hall problem: if there are 1,000 doors instead of three, and only one car and 999 goats, and Monty reveals 998 goats before the offer to switch etc.

The interesting thing to me is that what seems intuitively obvious varies a great deal between people of otherwise similar reasoning skills, which makes what’s counter-intuitive vary greatly on the basis of individual psychological factors.