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| − | ==Problem==
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| − | Pigs like to eat carrots. Suppose a pig randomly chooses 6 letters from the set <math>\{c,a,r,o,t\}.</math> Then, the pig randomly arranges the 6 letters to form a "word". If the 6 letters don't spell carrot, the pig gets frustrated and tries to spell it again (by rechoosing the 6 letters and respelling them). What is the expected number of tries it takes for the pig to spell "carrot"?
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| − | ==Solution==
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| − | We first find the chance of the pig spelling "carrot" correctly in one try.
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| − | ===Solution 1a===
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| − | First, out of the <math>5^6</math> ways to choose the letters, only <math>\frac{6!}{2}</math> of them have the same letters as the word carrot. Then, given that the pig has chosen the words correctly, only <math>1</math> out of the <math>\frac{6!}{2}</math> ways to spell the word correctly.
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| − | The probability is thus
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| − | <cmath>
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| − | \frac{\frac{6!}{2}}{5^6} \cdot \frac{1}{\frac{6!}{2}} = \frac{1}{5^6}
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| − | </cmath>
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| − |
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| − | ===Solution 1b===
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| − | Considering each letter position individually, it is equally likely to be any
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| − | of the <math>5</math> possible letters. Thus, for each letter in carrot there is a
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| − | <math>\frac{1}{5}</math> chance the pig spells the letter in that position correctly.
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| − | The answer is thus <math>\frac{1}{5^6}</math>.
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| − | Now let <math>x</math> be the expected number of turns required for the pig to
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| − | guess correctly.
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| − | We have that
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| − | <cmath>
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| − | x = 1 + \frac{5^6 - 1}{5^6} \cdot x
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| − | </cmath>
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| − | which implies that <math>x = \boxed{15625}</math>
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