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Current learning objective: Notation for the Binomial: \( B= \) Binomial Probability Distribution Function Question 14 Score: 0 of 4 points Using the Binomial distribution, if \( n=8 \) and \( p=0.6 \), find \( P(x=3) \) find \( P(x=3) \). Round your answer to 4 decimal places, if needed. \( P(x=3) \approx 0.1239 \)

Ask by Pollard Wilson. in the United States
Mar 15,2025

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\( P(x=3) \approx 0.1239 \)

Solution

We use the Binomial probability formula: \[ P(x) = \binom{n}{x} p^x (1-p)^{n-x} \] For \( n=8 \), \( p=0.6 \), and \( x=3 \): \[ P(x=3) = \binom{8}{3} (0.6)^3 (0.4)^{8-3} \] 1. Calculate the binomial coefficient: \[ \binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \] 2. Compute the powers: \[ (0.6)^3 = 0.216 \quad \text{and} \quad (0.4)^5 = 0.01024 \] 3. Multiply the results: \[ P(x=3) = 56 \times 0.216 \times 0.01024 \approx 0.1239 \]

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Did you know that the binomial distribution is incredibly useful in real-life scenarios? For example, companies often use it to predict the success rate of new product launches. If they know the probability that a customer will buy their product, they can estimate how many products they'll sell if they reach a certain number of customers. This helps in planning production and inventory, making their operations super efficient! Now let’s dive into a common mistake people make when working with binomial problems. A frequent blunder is failing to correctly identify \( n \) and \( p \). Ensure you double-check these values before jumping into calculations. Also, remember that the order of successes and failures matters if you're not using the formula correctly, so be mindful of that when working through problems!

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