Question
Determine if given events are all independent: 1. Owning a chihuahua and having an herb garden. 2. Driving a car and having anxiety. 3. Picking a flower from a vase to take home, then your sister picking a flower from the same vase Yes, because none of the events are connected to each other. Yes, because all these events do not depend on each other. No, because the events in \#3 rely on each other to happen. No, all the events depend on each other to happen. Question 14 In a group of 200 car buyers, 60 bought alarm systems, 50 bought heated seats, and 30 bought an alarm system and heated seats. If a car buyer chosen at random bought an alarm system, what is the probability they also bought heated seats? 25\% 40\%wis 50\% 20\% \( 2 \% \)
Ask by Murray Wang. in the United States
Jan 25,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
**Question 1:**
- **Answer:** No, because the events in #3 rely on each other to happen.
**Question 14:**
- **Answer:** 50%
Solution
Let's address each question one by one, providing clear explanations for the answers.
---
**Question 1: Determine if the given events are all independent**
**Events:**
1. Owning a chihuahua and having an herb garden.
2. Driving a car and having anxiety.
3. Picking a flower from a vase to take home, then your sister picking a flower from the same vase.
**Options:**
- Yes, because none of the events are connected to each other.
- Yes, because all these events do not depend on each other.
- No, because the events in #3 rely on each other to happen.
- No, all the events depend on each other to happen.
**Answer:** **No, because the events in #3 rely on each other to happen.**
**Explanation:**
- **Events 1 and 2:** These pairs of events (Owning a chihuahua & having an herb garden; Driving a car & having anxiety) are generally independent because one does not influence the probability of the other occurring.
- **Event 3:** Picking a flower from a vase and then your sister picking a flower from the same vase are **dependent events**. The outcome of the first event affects the probabilities in the second event (e.g., if there are limited flowers, the first pick reduces the number available for the second pick).
Since **Event 3** introduces a dependency, **all the events are not independent**.
---
**Question 14: Probability Calculation**
**Scenario:**
- Total car buyers: 200
- Bought alarm systems: 60
- Bought heated seats: 50
- Bought both alarm systems and heated seats: 30
**Question:** If a car buyer chosen at random bought an alarm system, what is the probability they also bought heated seats?
**Options:**
- 25%
- 40%
- 50%
- 20%
- 2%
**Answer:** **50%**
**Explanation:**
To find the probability that a car buyer who bought an alarm system also bought heated seats, use the formula for conditional probability:
\[
P(\text{Heated Seats} \,|\, \text{Alarm System}) = \frac{P(\text{Alarm System and Heated Seats})}{P(\text{Alarm System})}
\]
Plugging in the numbers:
\[
P(\text{Alarm System}) = \frac{60}{200} = 0.3
\]
\[
P(\text{Alarm System and Heated Seats}) = \frac{30}{200} = 0.15
\]
\[
P(\text{Heated Seats} \,|\, \text{Alarm System}) = \frac{0.15}{0.3} = 0.5 \text{ or } 50\%
\]
Therefore, there is a **50% probability** that a car buyer who bought an alarm system also bought heated seats.
---
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
Owning a chihuahua and having an herb garden may seem unrelated, but they can be connected by lifestyle choices or gardening preferences, which could sway independence. However, unrelated personal habits might make these events more independent than they appear at first glance. Now, moving on to probabilities! To solve the car buyer problem: You know that 30 bought both alarm systems and heated seats. Since there are 60 who bought alarm systems, the probability a randomly chosen car buyer who bought an alarm system also bought heated seats is 30/60, which simplifies to 50%. What a neat way to mix math and decision making!
