Problem 16 - Algebra and Trigonometry

Question

Combination Lock A safe containing \(\$ 1,000,000\) is locked with a combination lock. You pay \(\$ 1\) for one guess at the six-digit combination. If you open the lock, you get to keep the million dollars. What is your expectation?

Step-by-Step Solution

Answer

The expectation of a guess is 0.

Step 1: Determine the number of possible combinations

A six-digit combination can have each digit from 0-9, totaling 10 choices per digit. Thus, the total number of combinations is given by \( 10^6 \), which equals 1,000,000.

Step 2: Calculate probability of guessing correctly

The probability of guessing the correct combination on a single attempt is \( \frac{1}{1,000,000} \) because there's only one correct combination out of 1,000,000 possibilities.

Step 3: Calculate the expected value of one guess

The expected value \( E \) for one guess is calculated by multiplying the probability of winning by the prize and subtracting the cost of playing. This is expressed as \[ E = \left(\frac{1}{1,000,000}\right) \times 1,000,000 - 1 = 1 - 1 = 0. \]

Step 4: Conclusion

The expected value of a single guess is \(0\). This means on average, you neither gain nor lose money by making a guess.

Key Concepts

Combinations

A combination is a way of selecting items from a collection, such that the order of selection does not matter. In the case of our combination lock exercise, we want to find out how many different ways we can select the six-digit code. Here, each of the six positions in the combination can be any digit from 0 to 9, offering 10 different possibilities for each position.

To calculate the total number of possible combinations, we raise the number of options per position to the power of the total number of positions:

• Number of choices for each digit: 10 (digits 0 through 9)
• Total number of digits in the combination: 6

Therefore, the formula used here is: \[ 10^6 = 1,000,000 \]
This means there are 1,000,000 possible six-digit combinations for the lock.

Expected Value

The expected value is a fundamental concept in probability and statistics that provides a measure of the center of a random variable's distribution. Simply put, it represents the average outcome if the scenario were to be repeated many times.

In our exercise, we look at the expected value of one attempt at guessing the lock's combination. To determine this, you calculate:

• The value of winning, which in this case is the prize amount of 1,000,000 dollars.
• The probability of hitting that jackpot with one guess, which is very low—only \(\frac{1}{1,000,000}\).
• The cost of making one guess, which is \$1.

The formula for expected value \(E\) is therefore:\[ E = \left(\frac{1}{1,000,000}\right) \times 1,000,000 - 1 = 1 - 1 = 0 \]
This calculation shows that, on average, you would neither win nor lose money by making a single guess.

Probability Calculation

Probability is a way to measure the likelihood that an event will occur. It is quantified as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. In our context of guessing a combination, understanding probability helps you realize how likely you are to win that million dollars.

To determine the probability of guessing the right combination, consider:

• There is exactly one correct combination.
• There are 1,000,000 possible combinations, as discussed earlier.

Thus, the probability \(P\) of correctly guessing the combination in one try is:\[ P = \frac{1}{1,000,000} \]
Given these odds, it becomes clear that the probability of winning on a single attempt is extremely low. This numeric representation helps comprehend why such lottery-like scenarios often result in expectations of zero gains.