Question

A bag contains \(5\) red and \(7\) blue balls. If two balls are drawn at random, what is the probability that both are red?

• A. \( \dfrac{5}{66} \)
• B. \( \dfrac{10}{66} \)
• C. \( \dfrac{5}{33} \)
• D. \( \dfrac{25}{66} \)

Hint:

For drawing without replacement: \[ P(A \cap B) = P(A) \times P(B|A) \] Alternatively, use combinations: \[ P(\text{both red}) = \frac{\binom{5}{2}}{\binom{12}{2}} \]

Answer 1

The Correct Option is B

Concept:
When objects are drawn without replacement, the probability of successive events changes after each draw. If the total number of objects is \(n\), and \(r\) objects satisfy a condition, the probability can be computed using either:

• Multiplication rule of probability, or
• Combination formula

\[ P(\text{both red}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]
Step 1: Determine the total number of balls.
\[ \text{Total balls} = 5 + 7 = 12 \]
Step 2: Compute the probability using the multiplication rule.
Probability that the first ball is red: \[ P(R_1) = \frac{5}{12} \] After drawing one red ball, remaining red balls \(=4\) and total balls \(=11\). Probability that the second ball is red: \[ P(R_2|R_1) = \frac{4}{11} \]
Step 3: Find the joint probability.
\[ P(\text{both red}) = \frac{5}{12} \times \frac{4}{11} \] \[ = \frac{20}{132} \] \[ = \frac{10}{66} \] \[ \therefore \text{The probability that both balls are red is } \frac{10}{66}. \]