Suppose we throw a dice once.

Then, which one of the following is/are correct?

(A) The probability of getting a number greater than 4 is $\frac{1}{3}$

(B) The probability of getting a number greater than or equal to 4 is $\frac{1}{3}$

(C) The probability of getting a number less than or equal to 3 is $\frac{1}{2}$

(D) The probability of getting a number less than or equal to 6 is 1.

Choose the correct answer from the options given below:

The correct option is (A), (C) and (D) only.

To determine which statements are correct, let us analyze the outcomes of throwing a standard six-sided die once.

When a die is rolled, the sample space (the set of all possible outcomes) is:
S = {1, 2, 3, 4, 5, 6}
Therefore, the total number of possible outcomes is 6.

The probability of an event is given by the formula:
$\text{Probability}=\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$
Let us evaluate each statement one by one:

Statement (A): The probability of getting a number greater than 4 is $\frac{1}{3}$ .
The numbers on a die greater than 4 are {5, 6}. This gives 2 favorable outcomes.
The probability of this event is:
$\frac{2}{6}=\frac{1}{3}$
Thus, Statement (A) is correct.

Statement (B): The probability of getting a number greater than or equal to 4 is $\frac{1}{3}$ .
The numbers on a die greater than or equal to 4 are {4, 5, 6}. This gives 3 favorable outcomes.
The probability of this event is:
$\frac{3}{6}=\frac{1}{2}$
Thus, Statement (B) is incorrect.

Statement (C): The probability of getting a number less than or equal to 3 is $\frac{1}{2}$ .
The numbers on a die less than or equal to 3 are {1, 2, 3}. This gives 3 favorable outcomes.
The probability of this event is:
$\frac{3}{6}=\frac{1}{2}$
Thus, Statement (C) is correct.

Statement (D): The probability of getting a number less than or equal to 6 is 1.
All numbers on a standard die {1, 2, 3, 4, 5, 6} are less than or equal to 6. This gives 6 favorable outcomes.
The probability of this event is:
$\frac{6}{6}=1$
Thus, Statement (D) is correct.

Therefore, statements (A), (C), and (D) are correct, which matches the option (A), (C) and (D) only.