Probability of '1' Appearing as the Last Digit of a Random Number

In the context of generating random numbers, one common question that arises is: What is the probability that the last digit of a randomly generated number is '1'? This article will explore this exact scenario using various mathematical approaches and practical tests.

Mathematical Approach

Let N be a n-digit random natural number. Consider the following constraints:

The leading digit cannot be 0 The digits can be any number from 0 to 9 We are specifically interested in the last digit being 1

If N is represented as overline{α1α2...αn}, then each digit {α1 ... αn} can take on specific values. For instance:

α1 can be {1, 2, ..., 9} αi can be {0, 1, 2, ..., n-1} for i {2, 3, ..., n-1} αn 1

In the above situation, the number of possible combinations can be calculated as:

9 × 10n-2 P(N overline{α1α2...αn}: αn 1) 1 / [9 × 10n-2]

Alternatively, if we remove the restriction on the first digit being 1, the number of combinations is:

α1 can be {2, 3, ..., 9} αi can be {0, 2, ..., 9} for i {2, 3, ..., n-1} αn 1

The probability in this case is:

8 × 9n-2 P(N overline{α1α2...αn}: αn 1) 1 / [8 × 9n-2]

Practical Tests and Observations

Using a simple program that utilizes a basic random number generator, I performed several tests to observe the frequency of the last digit being '1' in randomly generated numbers.

Test 1: With only 10 or 100 random numbers, the results were not particularly even. As the number of samples increased, the distribution became more balanced.

Test 2: Similar to Test 1, the initial results with fewer samples were skewed. However, with larger sample sizes, the distribution improved significantly.

Test 3: The outcomes varied, but with larger sample sizes, the results showed a more even distribution.

These tests indicate that the quality of the random number generator, the seed value, and the number of samples all play significant roles in the observed distribution.

Interestingly, the results with the seed values tested were generally consistent, suggesting that the outcome is not highly sensitive to minor changes in the starting value.

Based on these observations, we can conclude that, on average, the probability of a randomly generated number ending in '1' is approximately 1 in 10.

P(N ends in 1) 1 / 10

Regardless of the specific random number generator used or the starting seed value, the long-term average frequency of the last digit being '1' converges to 1 in 10.