A bag contains 4 red, 5 blue and 3 green balls. If two balls are drawn at random from the bag, then which of the following statements are correct?

(A) The probability that both balls are red is $\frac{1}{11}$.

(B) The probability that one ball is red, and one ball is blue is $\frac{10}{33}$.

(C) The probability that both balls are blue is $\frac{5}{33}$.

(D) The probability that both balls are green is $\frac{5}{11}$.

Choose the correct answer from the options given below:

The bag contains:

$\text{Red balls} = 4$

$\text{Blue balls} = 5$

$\text{Green balls} = 3$

$\text{Total balls} = 12$

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When drawing 2 balls at random, we first need the total number of ways to select 2 balls from 12.

Using combinations:

$\binom{12}{2} = \frac{12 \times 11}{2} = 66$

This will be our denominator for all probability calculations.

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Checking Statement (A): The probability that both balls are red is $\frac{1}{11}$

Number of ways to choose 2 red balls from 4:

$\binom{4}{2} = \frac{4 \times 3}{2} = 6$

Probability:

$P(\text{both red}) = \frac{6}{66} = \frac{1}{11}$

Statement (A) is correct.

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Checking Statement (B): The probability that one ball is red and one ball is blue is $\frac{10}{33}$

Number of ways to choose 1 red from 4 and 1 blue from 5:

$\binom{4}{1} \times \binom{5}{1} = 4 \times 5 = 20$

Probability:

$P(\text{one red, one blue}) = \frac{20}{66} = \frac{10}{33}$

Statement (B) is correct.

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Checking Statement (C): The probability that both balls are blue is $\frac{5}{33}$

Number of ways to choose 2 blue balls from 5:

$\binom{5}{2} = \frac{5 \times 4}{2} = 10$

Probability:

$P(\text{both blue}) = \frac{10}{66} = \frac{5}{33}$

Statement (C) is correct.

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Checking Statement (D): The probability that both balls are green is $\frac{5}{11}$

Number of ways to choose 2 green balls from 3:

$\binom{3}{2} = \frac{3 \times 2}{2} = 3$

Probability:

$P(\text{both green}) = \frac{3}{66} = \frac{1}{22}$

The statement claims $\frac{5}{11}$, but the actual probability is $\frac{1}{22}$

Statement (D) is incorrect.

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Statements (A), (B), and (C) are correct.