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NDA 2 2024 Maths, question 113

Topic: Probability

Q113

A fair coin is tossed till two heads occur in succession. What is the probability that the number of tosses required is less than 66?

  1. (a) 5/645/64
  2. (b) 15/3215/32
  3. (c) 31/6431/64
  4. (d) 19/3219/32Correct answer

Solution.

Answer: (d) 19/3219/32

  1. “Less than 6” means HH occurs somewhere in the first five tosses.
  2. Count its complement: binary sequences of length n without adjacent H satisfy aₙ=aₙ₋₁+aₙ₋₂, with a₁=2,a₂=3.
  3. Thus a₅=13.
  4. Of 32 equally likely five-toss sequences, 32−13=19 have HH, giving 19/32.