Problem 69 - Algebra and Trigonometry

Question

Combination Lock 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. 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 is approximately 0.9375%.

Step 1: Calculate Total Possible Combinations

The lock has three numbers for the combination, each ranging from 1 to 40. Hence, the total number of possible combinations is calculated as the product of the possible numbers for each position: \[ 40 \times 40 \times 40 = 64,000 \] Therefore, there are 64,000 possible combinations.

Step 2: Determine Total Combinations Tried in One Hour

The student can try 10 combinations per minute. In one hour, which is 60 minutes, the total number of combinations she can try is: \[ 10 \times 60 = 600 \] Hence, she can try 600 combinations in one hour.

Step 3: Calculate Probability of Opening Lock Within One Hour

The probability of opening the lock within one hour is the number of combinations she can try in an hour divided by the total possible combinations: \[ \text{Probability} = \frac{600}{64,000} = \frac{3}{320} \approx 0.009375 \] Thus, the probability is approximately 0.009375 or 0.9375%.

Key Concepts

Combination Lock

A combination lock is a security device that requires a sequence of numbers or symbols to open it. This sequence, also known as a combination, must be entered in the correct order for the lock to release its mechanism. Typically, these locks are found on lockers, safes, or luggage.

The nature of a combination lock is determined by:

• The range of numbers or symbols available.
• The number of positions in the sequence (usually three or four).

In this exercise, our lock features three numbers, each ranging from 1 to 40. A combination is valid only if the correct sequence and order are used.
Understanding how to systematically try combinations without repeating is essential if the combination is forgotten, as trying every possible combination ensures that one will eventually find the right one.

Calculating Probability

Probability is a measure of the likelihood that a given event will occur. In this problem, we want to calculate the probability of successfully opening the combination lock within one hour.

The total number of possible outcomes (combinations) is 64,000, as calculated by multiplying the possible numbers for each position (40 per position). The student can attempt 600 combinations in one hour.

To find the probability of opening the lock, the formula used is:
\[\text{Probability} = \frac{\text{Number of successful outcomes}}{\text{Total number of possible outcomes}}\]So, for this lock:\[\text{Probability} = \frac{600}{64,000} = \frac{3}{320} \approx 0.009375\]This result represents a 0.9375% chance of success within the hour, meaning it's quite a low probability.

Combinatorics

Combinatorics, a branch of mathematics, focuses on counting, arranging, and finding patterns in sets. It is a vital tool in calculating probabilities, as it helps in determining the total number of possible outcomes or configurations.

In the context of this problem, combinatorics is used to determine the total number of potential combinations on the lock, which is key to solving the probability. For a three-number lock with each number ranging from 1 to 40, the calculation is simple multiplication:

• 1st position: 40 options
• 2nd position: 40 options
• 3rd position: 40 options

Using these options, the formula is:\[40 \times 40 \times 40 = 64,000\]Combinatorics allows us to see clearly how many combinations there are, enabling us to then apply this information to calculate probabilities effectively.