A five-digit number is written down at random. The probability that the number written is such that when the digits are put in the reversed order the new number is also a five-digit number equal to the original number

Options

A. 0.01
B. 0.9
C. 0.81
D. none of these

Answer

To solve the problem of finding the probability that a randomly written five-digit number remains the same when its digits are reversed, we can follow these steps:

### Step 1: Determine the total number of five-digit numbers.
A five-digit number can be represented as \(ABCDE\), where:
- \(A\) (the first digit) cannot be 0 (to ensure it is a five-digit number), so it can take values from 1 to 9 (9 choices).
- \(B\), \(C\), \(D\), and \(E\) can each take values from 0 to 9 (10 choices each).

Thus, the total number of five-digit numbers is calculated as:
\[ \text{Total five-digit numbers} = 9 \times 10 \times 10 \times 10 \times 10 = 9 \times 10^4 = 90000. \]

### Step 2: Determine the number of favorable outcomes.
For a five-digit number to be the same when reversed, it must be in the form \(ABCBA\). This means:
- \(A\) must be the same as the last digit \(A\),
- \(B\) must be the same as the fourth digit \(B\),
- \(C\) is the middle digit.

The choices for each digit are:
- \(A\) has 9 choices (1 to 9),
- \(B\) has 10 choices (0 to 9),
- \(C\) has 10 choices (0 to 9).

Thus, the number of favorable outcomes is:
\[ \text{Favorable outcomes} = 9 \times 10 \times 10 = 900. \]

### Step 3: Calculate the required probability.
The probability \(P\) that a randomly written five-digit number is the same when its digits are reversed is given by the ratio of the number of favorable outcomes to the total number of outcomes:
\[ P = \frac{\text{Favorable outcomes}}{\text{Total five-digit numbers}} = \frac{900}{90000} = \frac{1}{100} = 0.01. \]

### Conclusion
The probability that a randomly written five-digit number is equal to its reverse is \(0.01\).

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