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. Next, we use the deep concept of expected value to formalize the notion of an “average number” of flips before witnessing a certain event. 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. The key to resolving the coin paradox is to combine several mathematical concepts. 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. Probability is full of these paradoxes, which challenge human intuition. However, as we learned in this video, if two separate coins are used, then Wilbur suddenly has an advantage. 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. Using probability, this puzzle highlights a remarkable paradox.
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March 2023
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