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The problem in the image involves a binomial distribution. The task is to find the probability of getting 1 or fewer successes in a binomial experiment where the probability of success on a single trial is \( p = 0.37 \) and the number of trials is \( n = 10 \).

This means we need to calculate the cumulative probability of getting 0 or 1 successes.

The binomial probability formula is:

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

Where:
- \( P(X = k) \) is the probability of exactly \( k \) successes,
- \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \),
- \( p \) is the probability of success,
- \( n \) is the number of trials, and
- \( k \) is the number of successes (in this case, either 0 or 1).

We need to calculate:

1. \( P(X = 0) \)
2. \( P(X = 1) \)
3. Add the two probabilities to get \( P(X \leq 1) \).

Let me calculate that for you.The probability of obtaining 1 or fewer successes in this binomial experiment is approximately \( 0.068 \) (rounded to three decimal places).

Would you like further details on the calculations or have any additional questions?

Here are five related questions you might explore:
1. How does the probability change if the number of trials increases?
2. What is the probability of getting exactly 2 successes?
3. How would you calculate the expected number of successes in this experiment?
4. How does changing the probability of success (\( p \)) affect the overall outcome?
5. What is the variance of the number of successes in a binomial distribution?

**Tip**: Binomial probabilities can be quickly calculated using a calculator or software, which is especially useful when dealing with larger values of \( n \) or \( k \).

A binomial experiment with probability of success p = 0.37 and n = 10 trials is conducted. What is the probability that the experiment results in 1 or fewer successes?

Calculate the probability of getting 1 or fewer successes in a binomial experiment with p = 0.37 and n = 10 trials. Detailed step-by-step solution with cumulative probability and binomial distribution concepts. Ideal for high school students (Grades 10-12).

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