Given:

10% of all drivers arrested for DUI in Illinois are repeat offenders.
21 people arrested for DUI in Illinois are selected at random.
This follows a binomial distribution with n = 21 and p = 0.1.
find the right answer
b. To find the probability that at least one person is a repeat offender, we calculate:

To find the probability that **at least one person is a repeat offender**, we use the **complement rule**. The complement rule states that:

\[
P(\text{at least one repeat offender}) = 1 - P(\text{no repeat offenders})
\]

In this case, the probability of **no repeat offenders** can be calculated using the binomial distribution formula. The binomial distribution gives the probability of exactly \(k\) successes in \(n\) trials, which is:

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

Where:
- \(n = 21\) (number of trials or people selected),
- \(p = 0.1\) (probability of success, i.e., a repeat offender),
- \(k = 0\) (because we are interested in the probability of no repeat offenders).

For \(k = 0\), the formula simplifies to:

\[
P(0) = \binom{21}{0} (0.1)^0 (0.9)^{21} = (1) \times (1) \times (0.9)^{21}
\]

Now calculate \((0.9)^{21}\):

\[
P(0) = (0.9)^{21}
\]

Finally, the probability that **at least one person is a repeat offender** is:

\[
P(\text{at least one}) = 1 - P(0) = 1 - (0.9)^{21}
\]

Let's calculate this value:

\[
P(\text{at least one}) = 1 - (0.9)^{21} \approx 1 - 0.122
\]

\[
P(\text{at least one}) \approx 0.878
\]

Thus, the probability that at least one person is a repeat offender is approximately **0.878** or **87.8%**.

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Do you need further clarification on any step, or would you like more details on the binomial distribution?

Here are five related questions you might find useful:
1. What is the probability that exactly two people are repeat offenders in this scenario?
2. How would the probability change if the sample size were 30 instead of 21?
3. What is the expected number of repeat offenders in this group of 21 people?
4. What is the variance of the number of repeat offenders in this scenario?
5. How does the binomial distribution differ from the normal distribution in this case?

**Tip**: When calculating probabilities involving "at least one," using the complement rule can often simplify the process significantly.

Given: 10% of all drivers arrested for DUI in Illinois are repeat offenders. 21 people arrested for DUI in Illinois are selected at random. This follows a binomial distribution with n = 21 and p = 0.1. Find the probability that at least one person is a repeat offender.

Calculate the probability that at least one out of 21 randomly selected DUI arrestees in Illinois is a repeat offender using the binomial distribution and complement rule. Understand the core concepts of probability and the binomial theorem. Suitable for students in Grades 10-12.

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