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Chapter 5: Problem 39

Outside a home, there is a keypad that can be used to open the garage if the correct four-digit code is entered. (a) How many codes are possible? (b) What is the probability of entering the correct code on the first try, assuming that the owner doesn't remember the code?

Short Answer

Expert verified
There are 10,000 possible codes. The probability of entering the correct code on the first try is 0.0001.

Step by step solution

01

Title - Determine Total Possible Codes

First, note that each digit in the four-digit code can be any number from 0 to 9. This means there are 10 possible choices for each digit. To find the total number of possible codes, calculate the product of the number of choices for each digit: Total Possible Codes = 10 * 10 * 10 * 10.
02

Title - Calculate Total Possible Codes

Using the formula from Step 1, compute the total number of possible codes: \[ 10^4 = 10 \times 10 \times 10 \times 10 = 10000 \] Thus, there are 10,000 possible four-digit codes.
03

Title - Determine Probability Formula

The probability of guessing the correct code on the first try is the ratio of the number of successful outcomes (1 correct code) to the total number of possible outcomes (10,000 possible codes). Probability = \[ \frac{1 \text{ correct code}}{10000 \text{ total codes}} \]
04

Title - Calculate Probability

Using the formula from Step 3, calculate the probability: \[ \frac{1}{10000} = 0.0001 \] This means the probability of entering the correct code on the first try is 0.0001.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combinatorics
Combinatorics is a field of mathematics dealing with counting, arrangement, and combination of objects. In our problem, we use combinatorics to determine how many different four-digit codes are possible given certain constraints. Each digit can range from 0 to 9, making each position in the code have 10 different options. To calculate the total number of codes, we multiply the number of choices for each digit position together. Because there are four digits:
Probability Calculation
Probability helps us determine how likely an event is to occur. In this case, we want to know the likelihood of guessing a specific four-digit code on the first try. We calculate this by finding the ratio of successful outcomes (one specific correct code) to the total number of possible outcomes (all possible codes). So, if there are 10,000 possible codes, and only one of them is correct, the probability of guessing the correct code is:
Four-Digit Codes
A four-digit code is a sequence of four numbers, each ranging from 0 to 9. This type of code is often used for security systems such as keypads. The total number of unique combinations can be found by calculating 10 to the power of 4, because each digit has 10 different possibilities. This means there are 10,000 unique four-digit codes that can be created. When trying to guess one specific code out of these 10,000 options, it's essential to understand that the probability of guessing correctly is quite low due to the large number of possible combinations.

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Most popular questions from this chapter

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