Which of the following statements is/are correct .

Options

A. probability that the square of a randomly selected integer ends is 4 is `(2)/(5)`.
B. probability that the square of a randomlyselected integer ends in 4 is `(1)/(5)`.
C. probability that the `4^th` power of a randomly selected integer ends in 1 is `(2)/(5)`.
D. probability that square of a randomly selected integer ends in 3 or 7 is zero.

Answer

To determine which statements are correct regarding the probabilities of the last digits of squares and fourth powers of randomly selected integers, we will analyze each statement step by step.

### Step 1: Analyze the last digit of squares of integers
The last digit of a square of an integer can only be one of the following digits: 0, 1, 4, 5, 6, or 9.

- If the last digit of the integer is:
- 0 → square ends in 0
- 1 → square ends in 1
- 2 → square ends in 4
- 3 → square ends in 9
- 4 → square ends in 6
- 5 → square ends in 5
- 6 → square ends in 6
- 7 → square ends in 9
- 8 → square ends in 4
- 9 → square ends in 1

From this, we can see that the last digit of a square will be 4 only if the last digit of the integer is either 2 or 8.

### Step 2: Calculate the probability
The total number of possible last digits (0-9) is 10. The favorable outcomes for the last digit being 4 are 2 (from digits 2 and 8).

Thus, the probability \( P \) that the square of a randomly selected integer ends in 4 is:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{2}{10} = \frac{1}{5} \]

### Step 3: Evaluate the statements
- **Statement A:** The probability that the square of a randomly selected integer ends in 4 is \( \frac{2}{5} \). **(Incorrect)**
- **Statement B:** The probability that the square of a randomly selected integer ends in 4 is \( \frac{1}{5} \). **(Correct)**
- **Statement C:** The probability that the fourth power of a randomly selected integer ends in 1 is \( \frac{2}{5} \).

### Step 4: Analyze the last digit of fourth powers
The last digit of a fourth power of an integer can be determined similarly:
- If the last digit of the integer is:
- 0 → fourth power ends in 0
- 1 → fourth power ends in 1
- 2 → fourth power ends in 6
- 3 → fourth power ends in 1
- 4 → fourth power ends in 6
- 5 → fourth power ends in 5
- 6 → fourth power ends in 6
- 7 → fourth power ends in 1
- 8 → fourth power ends in 6
- 9 → fourth power ends in 1

From this, we see that the last digit of a fourth power will be 1 if the last digit of the integer is either 1, 3, 7, or 9.

### Step 5: Calculate the probability for fourth powers
The favorable outcomes for the last digit being 1 are 4 (from digits 1, 3, 7, and 9).

Thus, the probability \( P \) that the fourth power of a randomly selected integer ends in 1 is:
\[ P = \frac{4}{10} = \frac{2}{5} \]

- **Statement C:** The probability that the fourth power of a randomly selected integer ends in 1 is \( \frac{2}{5} \). **(Correct)**

### Step 6: Analyze the last digit of squares for 3 or 7
The last digit of a square can never be 3 or 7. The possible last digits of squares are 0, 1, 4, 5, 6, and 9.

- **Statement D:** The probability that the square of a randomly selected integer ends in 3 or 7 is 0. **(Correct)**

### Conclusion
The correct statements are:
- Statement B: Correct
- Statement C: Correct
- Statement D: Correct

### Final Answer
**Correct Statements:** B, C, D