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| − | == Personal info ==
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| − | Name: Daniel O'Connor
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| − | (full name: Daniel Verity O'Connor)
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| − | Location: Los Angeles
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| − | == Contributions ==
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| − | * Created [[Chain Rule]] article
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| − | * Created [[Fundamental Theorem of Calculus]] article
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| − | == Random Math Problems ==
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| − | The following problems are part of a discussion I'm having with someone I know. Please don't comment about them, and most importantly please don't correct any errors or give any hints about better solutions.
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| − | Suppose you have a fair six-sided die and you're going to roll the die again and again indefinitely. What's the expected number of rolls until a <math>1</math> comes up on the die?
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| − | The probability that it will take one roll is <math>\frac{1}{6} </math>.
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| − | The probability that it will take two rolls is <math>\left(\frac56 \right)\left(\frac16 \right) </math>.
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| − | The probability that it will take three rolls is <math>\left(\frac56 \right)^2 \left(\frac16 \right) </math>.
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| − | The probability that it will take four rolls is <math>\left(\frac56 \right)^3 \left(\frac16 \right) </math>.
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| − | And so on.
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| − | So, the expected number of rolls that it will take to get a <math>1</math> is:
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| − | <math>1\cdot \frac{1}{6} + 2\cdot \left(\frac56 \right)\left(\frac16 \right) + 3\cdot \left(\frac56 \right)^2 \left(\frac16 \right) + 4 \cdot \left(\frac56 \right)^3 \left(\frac16 \right) + \cdots</math>.
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| − | What's the sum of this infinite series? It looks kind of like an infinite geometric series, but not exactly. Factoring out a <math>\frac16</math> makes it look a bit more like an infinite geometric series:
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| − | <math>\left(\frac16 \right) \left(1 + 2\cdot \left(\frac56 \right) + 3\cdot \left(\frac56 \right)^2 + 4 \cdot \left(\frac56 \right)^3 + \cdots \right)</math>
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| − | This is similar to a geometric series, which we know how to sum. But we have those annoying factors of <math>2</math>, <math>3</math>, <math>4</math>, etc. to worry about. Maybe we could play around with the formula for a geometric series to get a formula for this series. The formula for a geometric series is:
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| − | <math>1 + r + r^2 + r^3 + r^4 + \cdots = \frac{1}{1-r}</math>.
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| − | Differentiating both sides of this with respect to <math>r</math>, we find:
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| − | <math>1 + 2r + 3r^2 + 4r^3 + \cdots = -(1-r)^{-2}(-1) = \frac{1}{(1-r)^2}</math>.
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| − | So, we find that
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| − | <math>\left(\frac16 \right) \left(1 + 2\cdot \left(\frac56 \right) + 3\cdot \left(\frac56 \right)^2 + 4 \cdot \left(\frac56 \right)^3 + \cdots \right) = \frac16 \frac{1}{(1-\frac56)^2} = \frac16 (36) = 6</math>.
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| − | Which seems like a very reasonable answer, given that the die has six sides.
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