Here's something that should make perfect sense, but most people don't believe. There's no such thing as a biased coin - only a biased toss, if there's a catch. No, not "but there's a catch," literally "if" there's a catch preventing the coin from bouncing or rolling and there is no rotational spin on the flat axis of the coin. The summary of the explanation can be found in Teaching Statisitics by Gelman and Nolan.
"But," you might say, "what if the coin's center of gravity is closer to the head side?" or "what if one edge is heavier?" or "what if it's slightly concave like a frisbee?" Still, the coin is not the source of any bias that results. As long as the coin does not bounce or roll after it lands, it has an equal chance of heads and tails (I've never observed a coin landing vertically on its edge but I won't completely rule it out!). Just think, a coin tossed with only "flipping" momentum spends 50% of it's time heads up, and 50% of it's time heads down. No bounce, no roll, no spin, no bias.
E.T. Jaynes is Professor of Physics (and, as it turns out teaches statistics to physics students) at Washington University in St. Louis. To illustrate, he used a pickle jar (to view this you need GSView for free or some other postscript file, *.ps, viewer), which is top-heavy and concave. Tossing it 100 times without spin, bounce or roll, he got results (p=0.54) that were not statistically different from 0.5 (z = 0.04/sqrt(0.5^2/100) = 0.8 --> P(z>0.8) = 0.424). He also tossed it to favor tails (allowing it to roll) and to favor heads (by appling rotational spin to it). When it was tossed in a way that made it roll, heads came up zero out of 100 trials, and when it was tossed with rotational spin, it came up heads 99 times.
So, the moral of the story is this. I've discussed before the notion of bias in flipping one versus two discs to determine the pull preceding an ultimate game. This tells us that the disc (uneven as it might be) is not biased unless the tosser applies bias. If the bias in the tosses is the same, then even is a dominant choice, even if the direction of the bias is unknown (but it should be known if you see the toss - will it spin or roll?). This can be proven as long as you know that the area of a square is more than the area of a rectangle with equal perimeter, i.e. it can be shown that p^2 + (1-p)^2 >= 2p(1-p), with equality only at p = 0.5. However, that point is now moot. If the tosser is not manipulating the toss, any single flip (heads or tails) and any double flip (odd or even) has "fair odds."
I'm convinced. Hope you are too.
"But," you might say, "what if the coin's center of gravity is closer to the head side?" or "what if one edge is heavier?" or "what if it's slightly concave like a frisbee?" Still, the coin is not the source of any bias that results. As long as the coin does not bounce or roll after it lands, it has an equal chance of heads and tails (I've never observed a coin landing vertically on its edge but I won't completely rule it out!). Just think, a coin tossed with only "flipping" momentum spends 50% of it's time heads up, and 50% of it's time heads down. No bounce, no roll, no spin, no bias.
E.T. Jaynes is Professor of Physics (and, as it turns out teaches statistics to physics students) at Washington University in St. Louis. To illustrate, he used a pickle jar (to view this you need GSView for free or some other postscript file, *.ps, viewer), which is top-heavy and concave. Tossing it 100 times without spin, bounce or roll, he got results (p=0.54) that were not statistically different from 0.5 (z = 0.04/sqrt(0.5^2/100) = 0.8 --> P(z>0.8) = 0.424). He also tossed it to favor tails (allowing it to roll) and to favor heads (by appling rotational spin to it). When it was tossed in a way that made it roll, heads came up zero out of 100 trials, and when it was tossed with rotational spin, it came up heads 99 times.
So, the moral of the story is this. I've discussed before the notion of bias in flipping one versus two discs to determine the pull preceding an ultimate game. This tells us that the disc (uneven as it might be) is not biased unless the tosser applies bias. If the bias in the tosses is the same, then even is a dominant choice, even if the direction of the bias is unknown (but it should be known if you see the toss - will it spin or roll?). This can be proven as long as you know that the area of a square is more than the area of a rectangle with equal perimeter, i.e. it can be shown that p^2 + (1-p)^2 >= 2p(1-p), with equality only at p = 0.5. However, that point is now moot. If the tosser is not manipulating the toss, any single flip (heads or tails) and any double flip (odd or even) has "fair odds."
I'm convinced. Hope you are too.
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