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Home » 2(1.05)^t = 3(1.03)^t - Groen Casting

2(1.05)^t = 3(1.03)^t - Groen Casting

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 #### 264Question: An epidemiologist models the spread of a virus where each infected person has a $\frac{1}{3}$ chance of transmitting it to a contact. What is the probability that exactly 2 out of 4 contacts become infected?

📌 Solution: The probability follows a binomial distribution with parameters $n = 4$ (trials) and $p = \frac{1}{3}$ (success probability). The formula for exactly $k$ successes is $\binom{n}{k} p^k (1-p)^{n-k}$. For $k = 2$:

📌 \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?

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