According to the U.S. Coast Guard, sixty-five percent of boating accidents occur in light wind condition. Suppose 20 boating accidents are randomly selected.

Find the probability that between 8 and 15 exclusive occurred in light wind conditions. answer in 4 decimals

We are dealing with a binomial distribution because there are a fixed number of trials (20 boating accidents), each trial has two possible outcomes (the accident occurred in light wind conditions or not), and the probability of success (light wind condition accident) is the same for each trial.

Let:

- \( n = 20 \) (number of trials),
- \( p = 0.65 \) (probability of success — an accident in light wind conditions),
- \( X \) be the random variable representing the number of accidents that occurred in light wind conditions.

We need to find the probability that the number of such accidents \( X \) is between 8 and 15 (exclusive), meaning \( 9 \leq X \leq 14 \).

The binomial probability formula is:

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

Where:

- \( \binom{n}{k} \) is the binomial coefficient.

To find the probability that \( X \) is between 9 and 14, we sum the individual probabilities for \( X = 9, 10, 11, 12, 13, 14 \).

Let me calculate this probability.The probability that between 9 and 14 boating accidents (exclusive) occurred in light wind conditions is approximately \( 0.7350 \).

Would you like further details or have any questions?

Here are five related questions to expand on this:

1. How would the probability change if only 10 boating accidents were selected instead of 20?
2. What if the probability of accidents in light wind conditions was 0.7 instead of 0.65?
3. How do you calculate the expected number of accidents in light wind conditions?
4. What is the probability that exactly 10 accidents occurred in light wind conditions?
5. How does the variance of a binomial distribution depend on the number of trials and probability?

**Tip:** The sum of probabilities in any binomial distribution across all possible outcomes (0 to \( n \)) will always equal 1.

According to the U.S. Coast Guard, sixty-five percent of boating accidents occur in light wind conditions. Suppose 20 boating accidents are randomly selected. Find the probability that between 8 and 15 exclusive occurred in light wind conditions. Answer in 4 decimals.

This problem explores how to use binomial distribution to calculate the probability of boating accidents in light wind conditions. Given that 65% of accidents occur in light wind, learn how to compute the probability for a range of accidents using binomial probabilities. Suitable for high school students in Grades 10-12.

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