Statistical Modeling, Causal Inference, and Social Science: The unicorn of probability theory

The unicorn of probability theory

"A coin with probability p > 0 of turning up heads is tossed . . . " -- Woodroofe, Probability with Applications (1975, p. 108)

"Suppose a coin having probability 0.7 of coming up heads
is tossed . . . " -- Ross, Introduction to Probability Models (2000, p. 82)

The biased coin is the unicorn of probability theory—-everybody has heard of it, but it has never been spotted in the flesh. As with the unicorn, you probably have some idea of what the biased coin looks like—-perhaps it is slightly lumpy, with a highly nonuniform distribution of weight. In fact, the biased coin does not exist, at least as far as flipping goes.

MDM

Actually, I think you're wrong. Consider a coin whose center of mass is close to the rim, and is flipped such that its axis of rotation is perpendicular to the line formed by the center of mass and the radius. I think that the momentum of the spin would carry it around so that, on landing, it would land on one side for more than 180 degrees of its rotation. That is, if the coin hits the (flat) hand on the weighted side, even if the angle is > 90 degrees, it would continue to rotate toward 0 degrees.

Andrew

Hmmm, maybe you're right! I'd still stand by my claim that that Ross's coin wiht Pr(heads)=0.7 doesn't exist anywhere. Continuing the unicorn analogy: it doesn't exist, but maybe it could be built...

Mike Anderson

So flip something non-homogeneous, like thumbtacks.

Mikhail

What about throwing knives? Properly balanced - i.e., handle-vs.-blade - they stick blade-first into their target even in the hands of an amateur. The knife is never allowed to bounce, so it is possble to bevel something so that its rotation through the air is affected.

Andrew

Mikhail,

I am certainly no expert on knife-throwing (!) but my guess is that what's going on here is that there are so few spins that a thrower can control it. Also, when throwing a knife, the thrower always starts by grabbing the handle, not the blade (or so I assume), and so he or she just has to toss it so that Pr(initial side lands first) is low. With coin flipping, the initial state is arbitrary, which is why there's no way to have Pr(heads)=0.7 (even if you can cleverly toss so that Pr(initial side lands first)=0.7, for example).

Maribel

I've seen an unicorn, or I think so. The Spanish 1 euro coin is biased. When you flip one of this coins, heads seem to have a bigger probability than tails. In fact, football referees don't use them as they are unbalanced!!!

Andrew

Maribel,

I'm skeptical. Try flipping the coin 100 times.

pm

I'm a 7th grader doing a science project. I've watched people checking out at the supermarket. The lines in the center always have the most people. My theory was that people would choose the shortest line and it would be random. This is not the case. Anyway I was wondering if you guys knew of any probability theory to predict what lines a group of people would choose.

thank you
jack

Joel

I was trying to find some info online and came up with nothing. Perhaps someone here can help. What I'm trying to find out is whether or not any common US coin (particularly a quarter) has an uneven weight distribution on either side. To state it differently: do the designs protruding from the metal on the heads side have the same weight as the those on the tails side? If they aren't exactly equal (and the coin is allowed to bounce off any surface until it lies still - as most people tend to do), then knowing which side weighs slightly more can give you a small advantage in determining the outcome (however miniscule it may be).

Kimberlee Gerritzen

Have any of you all done an experiment on whether or not you could predict the outcome when flipping a coin(a quarter)?

qwertyuiop

I disagree with MDM. any alteration to the centre of mass will be at such a small angle from the rim, it will effectively have no effect upon the rotation. And as you said, to have a chance of making a difference, you would have to spin it with incredible skill. A 10p coin, for instance, has a diameter of 24.5mm and a thickness of 1.5mm. therefore the angle you span it at would have to be arcsin 1.5/24.5 = 3.5 degrees. it is not possible to spin a coin sideways without effectively throwing it on the floor