For drivers aged 20-24 there is a 34% chance of having a car accident in a one year period (based on data from the National Safety Council).

(a)	Based on this data, in a group of 9 randomly selected drivers aged 20-24, find the probability that at least 2 of them will have a car accident in the next year.

This problem can be modeled using the **binomial distribution**, as we are dealing with a fixed number of trials (9 drivers), two possible outcomes for each trial (accident or no accident), and a constant probability of success (having an accident) for each driver.

### Step-by-Step Solution

Let’s define:
- \( n = 9 \) (number of trials, i.e., drivers),
- \( p = 0.34 \) (probability of having an accident for each driver),
- \( q = 1 - p = 0.66 \) (probability of not having an accident),
- We are asked to find the probability that **at least 2 drivers** have a car accident. That means we want \( P(X \geq 2) \), where \( X \) is the number of drivers that have accidents.

#### Binomial Probability Formula
The probability mass function of a binomial distribution is given by:
\[
P(X = k) = \binom{n}{k} p^k q^{n-k}
\]
Where:
- \( \binom{n}{k} \) is the binomial coefficient, representing the number of ways to choose \( k \) drivers from \( n \),
- \( p^k \) is the probability of \( k \) successes (accidents),
- \( q^{n-k} \) is the probability of \( n-k \) failures (no accidents).

To find \( P(X \geq 2) \), we use:
\[
P(X \geq 2) = 1 - P(X < 2) = 1 - [P(X = 0) + P(X = 1)]
\]
This simplifies the problem, as calculating \( P(X = 0) \) and \( P(X = 1) \) is easier than calculating all probabilities for \( X = 2, 3, \dots, 9 \).

#### Calculate \( P(X = 0) \) and \( P(X = 1) \):
1. **\( P(X = 0) \)**: No drivers have an accident.
\[
P(X = 0) = \binom{9}{0} (0.34)^0 (0.66)^9 = (1)(1)(0.66^9) = 0.66^9
\]

2. **\( P(X = 1) \)**: Exactly 1 driver has an accident.
\[
P(X = 1) = \binom{9}{1} (0.34)^1 (0.66)^8 = 9(0.34)(0.66^8)
\]

Let me compute these values.The probability that at least 2 of the 9 randomly selected drivers aged 20-24 will have a car accident in the next year is approximately **0.866** or **86.6%**.

Would you like further clarification or details? 

Here are 5 related questions to consider:
1. What is the probability that exactly 3 drivers have a car accident?
2. How does the probability change for different age groups?
3. How would the result change if the number of drivers was increased to 15?
4. What is the expected number of drivers to have an accident in a group of 9?
5. How can we use the binomial distribution to estimate accident probabilities for other risk factors?

**Tip:** The sum of probabilities for all possible outcomes in a binomial distribution must equal 1. This is a useful check for your calculations.

For drivers aged 20-24 there is a 34% chance of having a car accident in a one year period. Based on this data, in a group of 9 randomly selected drivers aged 20-24, find the probability that at least 2 of them will have a car accident in the next year.

Explore how to calculate the probability that at least 2 out of 9 drivers aged 20-24 will have a car accident in the next year using the binomial distribution. This detailed step-by-step solution includes key concepts like probability, binomial formulas, and combinatorics. Suitable for students in Grades 10-12.

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