Question: A linguist analyzes a hypothetical language where each word is 4 phonemes long, each phoneme independently from a set of 5. What is the probability that a randomly chosen word has exactly 2 distinct phonemes?
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- \binom{5}{2} = 10, \quad \left(\frac{1}{3}\right)^2 = \frac{1}{9}, \quad \left(\frac{2}{3}\right)^3 = \frac{8}{27}.
- Multiplying: $ 10 \cdot \frac{1}{9} \cdot \frac{8}{27} = \frac{80}{243} $.
- \boxed{\dfrac{80}{243}}
- Solution: First, choose 2 phonemes from 5: $ \binom{5}{2} = 10 $. For each pair, the number of 4-phoneme words using both phonemes (but not just one) is $ 2^4 - 2 = 14 $ (subtracting the two monochromatic words). Total favorable outcomes: $ 10 \cdot 14 = 140 $. Total possible words: $ 5^4 = 625 $. Probability:
- \frac{140}{625} = \frac{28}{125}.
- \boxed{\dfrac{28}{125}}
