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To determine the probability \( P(1 \text{ or fewer}) \) for a binomial experiment where the number of trials \( n = 8 \) and the success probability \( p = 0.3 \), we need to calculate:

\[ P(X \leq 1) = P(X=0) + P(X=1) \]

Where \( X \) is the number of successes. The probability mass function of a binomial distribution is given by:

\[ P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \]

1. For \( k = 0 \):
\[ P(X=0) = \binom{8}{0} \times 0.3^0 \times (1-0.3)^{8} = 1 \times 1 \times 0.7^8 \]

2. For \( k = 1 \):
\[ P(X=1) = \binom{8}{1} \times 0.3^1 \times (1-0.3)^{7} = 8 \times 0.3 \times 0.7^7 \]

Let's calculate these probabilities.The probability \( P(1 \text{ or fewer}) \) is approximately \( 0.2553 \).

Would you like to delve into more details on this calculation or do you have any further questions? Here are a few related topics you might find interesting:

1. How would the probability change if the success probability \( p \) was different?
2. What is the expected value and variance of this binomial distribution?
3. How to calculate the probability for more than 1 success, such as \( P(2 \text{ or more}) \)?
4. What are the general properties of a binomial distribution?
5. How to visualize the probability distribution of this binomial experiment?

**Tip**: Understanding binomial distributions can be enhanced by comparing theoretical calculations with simulation results, which can help in visualizing how the probabilities are distributed across different outcomes.

A binomial experiment has the given number of trials n and the given success probability p. n = 8, p = 0.3. Determine the probability P(1 or fewer). Round the answer to at least four decimal places.

Learn how to calculate the binomial probability P(1 or fewer) for a binomial experiment with 8 trials and a success probability of 0.3. This problem covers key concepts in probability theory and binomial distribution. Suitable for students in Grades 11-12.

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