A bag contains 6 white and 4 black balls. Two balls are drawn at random without replacement. What is the probability that both balls are of the same colour?

1. A

   1/10

2. B

   1/5

3. C

   2/5

4. D

   7/15

Appeared in: JIPMER Nursing Officer-2024

Explanation

• The problem requires calculating the probability of two mutually exclusive events: drawing two white balls OR drawing two black balls.
• The total probability is the sum of the individual probabilities for each case.
• Probability of drawing two white balls: P(W,W) = (6/10) * (5/9) = 30/90.
• Probability of drawing two black balls: P(B,B) = (4/10) * (3/9) = 12/90.
• Total Probability = P(W,W) + P(B,B) = 30/90 + 12/90 = 42/90, which simplifies to 7/15.

Why Other Options Were Wrong

• Option A: This value is incorrect and likely results from a calculation error, such as miscalculating the fractions or combinations.
• Option B: 1/5 is equal to 3/15. This is incorrect. It might arise from incorrectly calculating one of the probabilities, for example, only considering the probability of drawing two black balls (2/15) and then making a further error.
• Option C: 2/5 is equal to 6/15. This is incorrect. It might be a result of miscalculating the sum of probabilities or an error in simplifying the fractions.

Related Visual

• Visual 1: Flowchart: A probability tree diagram showing the two main branches (first ball is White or Black) and the subsequent second-level branches for the second draw, highlighting the paths for (W,W) and (B,B).

Clinical Relevance

• Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Probability of dependent events without replacement as background academic context rather than a clinical decision trigger.
• This is a quantitative aptitude question designed to test mathematical and logical reasoning skills.
• There is no direct clinical or nursing relevance for this specific probability problem.
• However, a strong foundation in mathematics is essential for nurses for tasks such as medication dosage calculation, interpreting lab results, and understanding statistical data in research.

How to Approach the Question

• First, identify the total number of items in the set (10 balls) and the number in each category (6 white, 4 black).
• Recognize that the draws are 'without replacement', meaning the total number of balls and the number in the chosen category decrease after the first draw.
• Break down the problem into the possible scenarios that satisfy the condition 'both balls are of the same colour': (1) both are white, (2) both are black.
• Calculate the probability for the first scenario (both white) by multiplying the probability of the first draw by the probability of the second draw.
• Calculate the probability for the second scenario (both black) in the same way.
• Since these two scenarios are mutually exclusive, add their probabilities together to find the total probability.

Concept Tested & Keywords

• Concept Tested: Probability of dependent events without replacement.
• Stem keywords: probability, 6 white balls, 4 black balls, two balls drawn, without replacement
• Lead-in keywords: probability, both balls are of the same colour

Question ID

QhTXUVjSkUhBKLDIWAgBQW

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