It is estimated that 78% of drivers use their turn signal.

Part A: What is the probability that exactly 3 drivers use their turn signal if a police officer pulls over five drivers? (5 points)

Part B: What is the probability the next driver using their turn signal that the police officer pulls over is the fifth driver? (5 points) (10 points)

We are dealing with a **binomial probability distribution** in Part A and a **geometric probability distribution** in Part B. Let's go step by step.

### Part A: Probability that exactly 3 out of 5 drivers use their turn signal

This follows a **binomial distribution**, where:
- \( n = 5 \) (the number of trials, i.e., drivers),
- \( p = 0.78 \) (the probability that a driver uses their turn signal),
- \( q = 1 - p = 0.22 \) (the probability that a driver does **not** use their turn signal),
- \( k = 3 \) (the number of successes, i.e., the number of drivers using their turn signal).

The binomial probability formula is:

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

Where \( \binom{n}{k} \) is the binomial coefficient (combination formula):

\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]

Now let's calculate \( P(X = 3) \):

\[
P(X = 3) = \binom{5}{3} (0.78)^3 (0.22)^2
\]

First, calculate the binomial coefficient \( \binom{5}{3} \):

\[
\binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10
\]

Now plug everything into the formula:

\[
P(X = 3) = 10 \times (0.78)^3 \times (0.22)^2
\]

Let's calculate this.

### Part B: Probability the next driver using their turn signal is the fifth driver

This follows a **geometric distribution**, where:
- \( p = 0.78 \) (the probability that a driver uses their turn signal),
- \( q = 0.22 \) (the probability that a driver does **not** use their turn signal).

The probability of the first success happening on the \( n \)-th trial (in this case, the 5th driver) in a geometric distribution is given by:

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

For the 5th driver:

\[
P(X = 5) = (0.22)^4 \times 0.78
\]

Now, let's calculate these probabilities.### Part A:
The probability that exactly 3 out of 5 drivers use their turn signal is approximately **0.2297** or **22.97%**.

### Part B:
The probability that the next driver using their turn signal is the fifth driver is approximately **0.0018** or **0.18%**.

Would you like more details or have any questions?

Here are 5 related questions for deeper understanding:
1. How would the probability change if 7 drivers were pulled over instead of 5 in Part A?
2. How would the geometric distribution formula apply if the 4th driver was being considered instead of the 5th in Part B?
3. What happens to the probabilities if the rate of drivers using their turn signal decreases to 60%?
4. How can you use cumulative binomial probabilities to calculate the likelihood of fewer than 3 drivers using their turn signal?
5. How is a binomial distribution different from a normal distribution?

**Tip:** The binomial distribution is ideal for modeling scenarios with a fixed number of independent trials, each with the same probability of success (like pulling over a certain number of drivers).

It is estimated that 78% of drivers use their turn signal.

Part A: What is the probability that exactly 3 drivers use their turn signal if a police officer pulls over five drivers?

Part B: What is the probability the next driver using their turn signal that the police officer pulls over is the fifth driver?

Calculate the probability of drivers using turn signals with binomial and geometric probability distributions. Find out the chances that exactly 3 out of 5 drivers use their signal, and the probability the 5th driver is the first to use a signal. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation