Geometric with multiple success milestones

Questions requiring geometric distribution for various milestones of first success (exactly nth trial, before nth trial, after nth trial) and may include finding second success timing, all within the same continuing scenario without switching to a fixed-trial binomial setup.

2 questions

OCR S1 2015 June Q5
5 Each year Jack enters a ballot for a concert ticket. The probability that Jack will win a ticket in any particular year is 0.27 .
  1. Find the probability that the first time Jack wins a ticket is
    (a) on his 8th attempt,
    (b) after his 8th attempt.
  2. Write down an expression for the probability that Jack wins a ticket on exactly 2 of his first 8 attempts, and evaluate this expression.
  3. Find the probability that Jack wins his 3rd ticket on his 9th attempt and his 4th ticket on his 12th attempt.
OCR S1 2013 January Q8
8 Sandra makes repeated, independent attempts to hit a target. On each attempt, the probability that she succeeds is 0.1 .
  1. Find the probability that
    (a) the first time she succeeds is on her 5th attempt,
    (b) the first time she succeeds is after her 5th attempt,
    (c) the second time she succeeds is before her 4th attempt. Jill also makes repeated attempts to hit the target. Each attempt of either Jill or Sandra is independent. Each time that Jill attempts to hit the target, the probability that she succeeds is 0.2 . Sandra and Jill take turns attempting to hit the target, with Sandra going first.
  2. Find the probability that the first person to hit the target is Sandra, on her
    (a) 2nd attempt,
    (b) 10th attempt. \href{http://physicsandmathstutor.com}{physicsandmathstutor.com}
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