Probability of Last Four Digits of a Phone Number Being Even: Explained

When dealing with the probability that the last four digits of a phone number are even, we must take into account the fact that phone numbers can vary widely across different countries. This article will explore the probability in a general sense, focusing on the United States, and provide a clear explanation of the calculation.

Understanding the Problem

The question posed by the user is crucial because phone numbers are structured differently across countries. For the sake of simplicity, let’s assume a standard 10-digit phone number format, which is common in the United States. A phone number would look like this: 123-456-7890. The last four digits would be 7890.

Calculating the Probability

Each digit of a phone number can be any number from 0 to 9. This means that each digit has 10 possible values. Out of these 10 possible values, 5 are even numbers (0, 2, 4, 6, 8). Therefore, the probability that a single digit is even is 1/2 or 0.5.

Now, we need to find the probability that the last four digits of a phone number are all even. Since each digit is independent of the others, we can simply multiply the probabilities.
[text{Probability of one digit being even} frac{1}{2}] [text{Probability of all four digits being even} left(frac{1}{2}right)^4 frac{1}{16} 0.0625]

Expected Probability

The term ‘expected probability’ can refer to the average or predicted probability over multiple trials of the same event. In the case of a phone number, the probability that the last four digits are even is not expected to change based on the number of trials due to the independence of individual digits. Thus, the expected probability is simply the calculated probability of 0.0625 or 6.25%.

Why the Example Given May Be Incorrect

The statement 'S1/2^4 1/16' is correct, but it should be interpreted carefully. The notation 'SN' is not clear, so it may have been a typo. The correct calculation is 0.5^4, which equals 0.0625. The author may have meant to say 'S', which could be a placeholder for 1, but it was not specified. Therefore, the provided solution is accurate if interpreted correctly as 0.5^4 1/16 0.0625.

Conclusion

In summary, the probability that the last four digits of a 10-digit phone number (like a U.S. phone number) are all even is approximately 6.25%. This probability remains consistent regardless of the specific phone number, given that the digits are randomly and uniformly distributed. Understanding and calculating such probabilities can be helpful in various scenarios, from cryptography to statistical analysis.