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Chapter 4: Problem 13

Express all probabilities as fractions.The author owns a safe in which he stores all of his great ideas for the next edition of this book. The safe "combination" consists of four numbers between 0 and 99, and the safe is designed so that numbers can be repeated. If another author breaks in and tries to steal these ideas, what is the probability that he or she will get the correct combination on the first attempt? Assume that the numbers are randomly selected. Given the number of possibilities, does it seem feasible to try opening the safe by making random guesses for the combination?

Short Answer

Expert verified
The probability of guessing the correct combination is \( \frac{1}{10{,}000{,}000} \). Random guessing is not feasible.

Step by step solution

01

Determine the total number of different possible combinations

Each number in the combination can be any number between 0 and 99. Since there are four numbers in the combination, and each number can be selected independently, the total number of combinations can be calculated using 100 choices for each of the four positions. Thus, the total number of possible combinations is given by\[ 100 \times 100 \times 100 \times 100 = 100^4 = 10{,}000{,}000. \]
02

Identify the number of successful outcomes

For the author trying to guess the combination, there is only one specific correct combination out of the 10,000,000 possibilities.
03

Calculate the probability

The probability of guessing the correct combination on the first attempt is the ratio of the number of successful outcomes to the total number of possible outcomes. Therefore, the probability is given by:\[ P(\text{correct combination}) = \frac{1}{10{,}000{,}000}. \]
04

Assess the feasibility

Given that there are 10,000,000 possible combinations and only one correct combination, the very low probability of guessing the combination correctly on the first attempt (\( \frac{1}{10{,}000{,}000} \)) suggests that it is not feasible to open the safe by making random guesses.

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Key Concepts

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

Combinatorics
Combinatorics is a branch of mathematics that deals with counting and arranging objects. In this exercise, we used combinatorial principles to calculate the total number of possible combinations for a safe's lock. If you have a lock where each position can take any number from 0 to 99, and there are four positions, you use the rule of product to find the total number of combinations. Each position has 100 possible choices, so the total number of combinations is calculated by multiplying the number of choices for every position: \[ 100 \times 100 \times 100 \times 100 = 100^4 = 10,000,000. \]Understanding combinatorics helps you figure out how large the possibility space is, which is essential for calculating probabilities.
Independent Events
In probability theory, events are called independent if the outcome of one event does not affect the outcome of another. In the context of the safe combination problem, each number in the combination is chosen independently from the others. This means that the choice of one number does not limit or influence the possible choices for another number.Why is this important? Because it allows us to multiply the number of choices for each position to get the total number of combinations. If the choices were not independent, we would need a different approach to calculate the total combinations. Understanding this independence simplifies our calculations and helps us grasp the basic principles of probability.
Probability Calculation
Calculating probability involves understanding the likelihood of an event happening out of the total number of possible outcomes. In our exercise, we needed to find the probability that another author guesses the correct combination of the safe on the first try.First, we identified the total number of possible combinations, which is 10,000,000. Then, we noted that there is only one correct combination. Probability is the ratio of the number of successful outcomes to the total number of possible outcomes. Hence, the probability of guessing the correct combination on the first attempt is:\[ P(\text{correct combination}) = \frac{1}{10,000,000}. \]This extremely low probability indicates that guessing the combination randomly is very unlikely to succeed.
Outcomes
Outcomes are the possible results of an event or experiment. For the safe combination problem, each possible sequence of four numbers is an outcome. Given that every number can range from 0 to 99, the total outcomes are all the combinations that can be formed with these numbers for four positions: \[ 100^4 = 10,000,000. \]Understanding outcomes helps in calculating probabilities. If outcomes are well-defined and measurable, it's easier to calculate the probability of a specific event happening. In our problem, there is only one successful outcome: the correct combination of 4 specific numbers. Knowing the number of possible outcomes lets us determine how rare鈥攐r frequent鈥� a specific outcome is. In this case, having only one correct combination among 10,000,000 outcomes shows the challenge of guessing it correctly.

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Most popular questions from this chapter

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