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Home » Expected pink = 400 × 0,5 = 200 - 500apps

Expected pink = 400 × 0,5 = 200 - 500apps

📅 March 13, 2026 👤 scraface

Mar 13, 2026

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📌 \binom{4}{2} \left(\frac{1}{3}\right)^2 \left(\frac{2}{3}\right)^2 = 6 \cdot \frac{1}{9} \cdot \frac{4}{9} = \frac{24}{81} = \frac{8}{27}.

📌 The probability is $\boxed{\dfrac{8}{27}}$.

📌 Question: A neural interface system processes 5 distinct signals using 3 unique channels. How many ways can the signals be assigned to channels if each channel can process any number of signals?

📌 Solution: Each of the 5 distinct signals has 3 independent choices (channels). By the multiplication principle, the total number of assignments is $3^5$. Calculating:

📌 The number of ways is $\boxed{243}$.

📌 Question: A sustainable agriculture startup divides 6 distinct crop types into 2 identical storage units. How many ways can the crops be partitioned if each unit must contain at least one type?

📌 Solution: This is equivalent to finding the number of ways to partition 6 distinct elements into 2 non-empty subsets, which is the Stirling number of the second kind $S(6, 2)$. The formula for $S(n, 2)$ is $2^{n-1} - 1$. For $n = 6$:

📌 2^{5} - 1 = 32 - 1 = 31.

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Number of distinct permutations of {A,A,B,C}: $ \frac{4!}{2!} = 12 $
Total: $4 \times 3 \times 12 = 144$ — correct.
Total outcomes: $4^4 = 256$ — correct.
So probability is $ \frac{144}{256} = \frac{9}{16} $? But $144/256 = 5...
Wait — but $ \frac{144}{256} = \frac{144 \div 16}{256 \div 16} = \frac...
But $144/256 = 0.5625$, and $9/16 = 0.5625$, correct.
But let’s double-check total favorable: is there overcount?
No — we fix the repeated word, then choose two others, then arrange. E...
Alternatively, compute using multinomial:
Number of sequences with one word once, one word twice, and a third wo...

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