The access code for a 

garage doorgarage door 

consists of three digits. Each digit can be any number from 

00 

through 

55​, 

and each digit can be repeated. Complete parts​ (a) and​ (b).

​(a) What is the probability of randomly selecting the correct access code on the first​ try?

​(b) What is the probability of not selecting the correct access code on the first​ try?




Let's analyze and solve both parts of this problem:

### Given Information:
- The access code consists of **three digits**.
- Each digit can be any number from **0** through **5**.
- Digits **can be repeated**.

### Step-by-Step Solution:

#### (a) Probability of randomly selecting the correct access code on the first try:

1. **Total possible codes**:
   Each digit can be **0, 1, 2, 3, 4, or 5** — giving **6** possible choices per digit.
   Since there are **3 digits**, the total number of possible codes is:
   \[
   6 \times 6 \times 6 = 216
   \]

2. **Number of correct codes**:
   There is only **1 correct code**.

3. **Probability**:
   The probability \( P \) of selecting the correct code on the first try is:
   \[
   P = \frac{\text{Number of correct codes}}{\text{Total possible codes}} = \frac{1}{216}
   \]

#### (b) Probability of not selecting the correct access code on the first try:

1. **Number of incorrect codes**:
   Since there are 216 possible codes and only 1 is correct, the number of incorrect codes is:
   \[
   216 - 1 = 215
   \]

2. **Probability**:
   The probability \( P \) of not selecting the correct code on the first try is:
   \[
   P = \frac{\text{Number of incorrect codes}}{\text{Total possible codes}} = \frac{215}{216}
   \]

### Final Answers:
- **(a)** Probability of selecting the correct code: \(\frac{1}{216} \approx 0.00463\) (approximately 0.463%)
- **(b)** Probability of not selecting the correct code: \(\frac{215}{216} \approx 0.99537\) (approximately 99.537%)

Would you like a deeper explanation or have any other questions?

---

Here are 5 related questions that expand on this topic:
1. What would the probability be if there were 4 digits instead of 3, with each digit ranging from 0 to 9?
2. How does the probability change if digits cannot be repeated in the access code?
3. What is the probability of randomly selecting a code with all three digits being the same (e.g., 000, 111)?
4. If you try 5 different codes at random, what is the probability that one of them is correct?
5. What is the probability of selecting a code with no zeros (only using digits 1 through 5)?

**Tip:** When calculating probabilities, always ensure the total possible outcomes and the favorable outcomes are correctly counted. This helps in setting up the problem correctly from the start.

The access code for a garage door consists of three digits. Each digit can be any number from 0 through 5, and each digit can be repeated. Complete parts​ (a) and (b). (a) What is the probability of randomly selecting the correct access code on the first​ try? (b) What is the probability of not selecting the correct access code on the first​ try?

Explore the probability of selecting the correct 3-digit garage door access code when each digit ranges from 0 to 5 and can be repeated. This problem involves combinatorics and basic probability concepts, perfect for Grades 7-10 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation