Using probability, this puzzle highlights a remarkable paradox. If Orville and Wilbur are flipping a single coin, with Orville winning as soon as heads is immediately followed by heads, and Wilbur as soon as heads is immediately followed by tails, then both have equal chances of winning. However, as we learned in this video, if two separate coins are used, then Wilbur suddenly has an advantage. Probability is full of these paradoxes, which challenge human intuition. These paradoxes are great examples of the value and power of mathematics: to identify and explain the truth when there are gaps in our natural intuition.

The key to resolving the coin paradox is to combine several mathematical concepts. Our starting point is to model a fair coin as a sequence of independent outcomes, each of which has 50% probability of coming up heads or tails. Next, we use the deep concept of expected value to formalize the notion of an “average number” of flips before witnessing a certain event. In the video, we intentionally used the more colloquial word “average” to ease intuition, as it sounds plausible that if a random variable X has an average value of A and a random variable Y has an average value of B, then the average value of the random variable (X+Y) is A+B. This property was at the heart of all of our analysis, and is often known as “linearity of expectation." That allowed us to write equations in terms of the expected values as variables, which we then solved with algebraic techniques. For further investigation into an even more complex situation with three consecutive coin flips, there is an engaging video by Numberphile, which introduces Penney’s Game.

Indeed, some of the most interesting solutions in mathematics (and in the world) rely on fluency across multiple mathematical topics, combined with creative problem solving in the face of challenge. Po-Shen Loh is on a mission to introduce this fluency through expii.com, a free, personalized math (and science) learning platform, which invites the world to share its knowledge. Don't forget to check out Expii's YouTube channel, too!

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