Problem 59 - Algebra and Trigonometry

Question

A student has locked her locker with a combination lock, showing numbers from 1 to 40, but she has forgotten the three-number combination that opens the lock. In order to open the lock, she decides to try all possible combinations. If she can try ten different combinations every minute, what is the probability that she will open the lock within one hour?

Step-by-Step Solution

Answer

The probability she will open the lock within one hour is \( \frac{3}{320} \) or approximately 0.9375%.

Step 1: Understanding the Problem

The problem involves finding the probability that a student will open her locker within a given timeframe by trying all possible combinations. We need to determine the total number of possible combinations for a lock with three numbers, each ranging from 1 to 40.

Step 2: Calculating Total Combinations

A combination lock with numbers ranging from 1 to 40 and using three numbers can have its numbers repeated. Thus, the total number of combinations is calculated by multiplying the number of options for each position: \[ 40 \times 40 \times 40 = 64,000 \] Therefore, there are 64,000 possible combinations.

Step 3: Determine Combinations Tried in One Hour

The student tries ten combinations every minute. To find out how many combinations she can try in one hour (60 minutes), we calculate: \[ 10 \times 60 = 600 \] Hence, she can try 600 combinations in one hour.

Step 4: Calculating the Probability

The probability that she will open the locker within one hour is the ratio of the number of combinations she can try in one hour to the total number of possible combinations. Therefore: \[ \text{Probability} = \frac{600}{64,000} = \frac{3}{320} \]

Step 5: Simplifying the Probability

Simplifying the fraction \( \frac{3}{320} \):This fraction cannot be reduced further, so \( \frac{3}{320} \) remains as is. This value, expressed as a percentage, is approximately 0.9375%. Thus, there is a very small chance she will open the lock within an hour.

Key Concepts

Combination Lock

A combination lock is a security device that requires a specific sequence of numbers to disengage the lock mechanism. These locks are commonly used on school lockers and safes. They ensure that only those with the correct combination can access the contents secured by the lock.

In this scenario, the lock uses three numbers ranging between 1 and 40. Importantly, each number in the sequence can repeat, meaning the lock could potentially be set to the combination (5-5-5), as well as (1-2-3). This allows for an extensive range of possible sequences, making the lock more secure against random guesses.

Understanding how combination locks function is pivotal for tackling problems involving them, especially when employing probability to predict successful access times.

Probability Calculation

Probability is a mathematical concept that measures the likelihood of a particular outcome. In this case, the student wants to know the probability of successfully opening her combination lock within an hour of attempts.

To calculate this probability, we use a simple formula: the number of successful outcomes (combinations tried within an hour) divided by the total number of possible outcomes (all possible lock combinations).

This calculation helps understand the odds of achieving a desired result in uncertain circumstances, a skill very useful in both academic exercises and real-world scenarios.

Lock Combinations

When determining the number of potential lock combinations, it's important to understand the principles of multiplication in probability. Here, we have a three-digit lock with each digit ranging from 1 to 40.

Therefore, for each of the three positions, there are 40 possible choices, independent from the others. Thus, the total number of combinations is obtained by multiplying the options for each individual position:

• 40 choices for the first number
• 40 choices for the second number
• 40 choices for the third number

This results in a total of: \[ 40 \times 40 \times 40 = 64,000 \] This means there are 64,000 possible different combinations, showing just how challenging it is to guess the right one purely by chance.

Repeated Numbers

In probability and combinations, the concept of repeated numbers plays a crucial role. Repeated numbers mean that each digit in a sequence can be the same as another. For example, in a combination lock from 1 to 40, using three numbers, (2-2-2) is just as valid as (3-5-7).

This repetition significantly increases the number of possible combinations, adding layers of difficulty in guessing the correct sequence using random attempts.

When repeats are allowed, it often requires more time or trials to uncover the correct sequence, illustrating how repeated numbers affect both security and probability calculations. Understanding this aspect helps in creating strategies to deal with locks and similar problems efficiently.