GATE CS 2026 Set 2 — Question 63
Go beyond PYQs with Success TrackerAI-powered personalised practice and doubt support. Unlimited practice on eligible plans; AI usage limits apply.NAT+2 / -0HardConditional ProbabilityProbability & StatisticsEngineering Mathematics
Engineering Mathematics → Probability & Statistics → Conditional Probability
Last updated
Question
Suppose an unbiased coin is tossed 6 times. Each coin toss is independent of all previous coin tosses. Let be the event that among the second, fourth, and sixth coin tosses, there are at least two heads. Let be the event that among the first, second, third, and fifth coin tosses, there are equal number of heads and tails.The conditional probability is equal to ____________. (rounded off to one decimal place)
Correct answer
0.5 to 0.5
Solution
We need to find .Let the outcomes of the 6 tosses be .
Number of ways to arrange 2H and 2T in 4 positions is . The other 2 tosses () can be anything ( ways).
.Calculate :
We split cases based on since it is common to both sets.Case 1:
- involves tosses . Condition: Equal heads and tails (2H, 2T).
- involves tosses . Condition: At least 2 heads.
Number of ways to arrange 2H and 2T in 4 positions is . The other 2 tosses () can be anything ( ways).
.Calculate :
We split cases based on since it is common to both sets.Case 1:
- For (tosses 1, 2, 3, 5): We have H at . We need 1H and 2T in . Ways = .
- For (tosses 2, 4, 6): We have H at . We need at least 1 more H in . Outcomes for are HH, HT, TH, TT. Valid are HH, HT, TH. Ways = 3.
- Total for Case 1: .
- For (tosses 1, 2, 3, 5): We have T at . We need 2H and 1T in . Ways = .
- For (tosses 2, 4, 6): We have T at . We need at least 2 more H in . Only HH is valid. Ways = 1.
- Total for Case 2: .
Continue learning with Success Tracker
A step still unclear? Work through it with support
Use Success Tracker to ask about the reasoning, then try another GATE CS question to check your understanding.
AI-powered practice· Unlimited practice on eligible plans
- PYQs with solutions
- Attempt available previous-year questions, then compare your reasoning with the worked solution. Coverage varies by stream.
- Practice that adapts
- Choose a topic, work on weaker areas and bookmark questions to revisit. Your attempts feed your progress tracking.
- AI doubt support
- Ask follow-up questions about a step or concept while practising, instead of stopping at the final answer.
Unlimited practice is available on eligible plans. Free practice and AI usage have limits; check the current plan allowances before choosing.
This page stays readable without an account. AI responses can be wrong; check them against the solution and source material.
More questions on Probability & Statistics
2026 Set 2 Q3A day can only be cloudy or sunny. The probability of a day being cloudy is , independent of…2026 Set 2 Q5‘When it is raining, peacocks dance.’ Based only on this sentence, which one of the following…2026 Set 2 Q8Figures (i) and (ii) represent intercity highway systems. The black dots represent cities and the…2026 Set 1 Q10An unbiased six-faced dice whose faces are marked with numbers 1, 2, 3, 4, 5, and 6 is rolled twice…2026 Set 2 Q10An unbiased six-faced dice whose faces are marked with numbers 1, 2, 3, 4, 5, and 6 is rolled twice…