Basic probability without replacement: A box has 4 white and 3 red balls. Two balls are drawn successively without replacement. What is the probability that both are white?
Electronics and Communication Engineering
Signals and Systems
Difficulty: Easy
Choose an option
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A2/7
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B4/21
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C3/7
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D8/21
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E1/7
Answer
Correct Answer: 2/7
Explanation
Introduction / Context:Sampling without replacement is a fundamental probability model. Here, drawing two balls successively from a small urn illustrates conditional probability and the multiplication rule.
Given Data / Assumptions:
- Total balls: 7 (4 white, 3 red).
- Two draws in succession without replacement.
- Event of interest: both balls are white.
Concept / Approach:
Use the multiplication rule: P(A ∩ B) = P(A) * P(B | A). Let A be “first ball white” and B be “second ball white given first white”.
Step-by-Step Solution:
P(first white) = 4/7.After removing one white, remaining white = 3, remaining total = 6.P(second white | first white) = 3/6 = 1/2.So, P(both white) = (4/7) * (3/6) = 12/42 = 2/7.Verification / Alternative check:
Combinatorial approach: favorable outcomes = C(4,2) = 6; total 2-ball selections = C(7,2) = 21; so probability = 6/21 = 2/7. Matches conditional method.
Why Other Options Are Wrong:
- 4/21 and 8/21 are common arithmetic slips; 3/7 counts single white outcomes incorrectly; 1/7 is too low.
Common Pitfalls:
- Accidentally using replacement or forgetting to update counts on the second draw.
- Confusing ordered draws with unordered combinations.
Final Answer:
2/7