r-th success on trial n

Find the probability that the r-th success (r ≥ 2) occurs on trial n (negative binomial distribution).

2 questions

OCR MEI Further Statistics Minor 2022 June Q4
4 Alex is practising bowling at a cricket wicket. Every time she bowls a ball, she has a \(30 \%\) chance of hitting the wicket.
  1. Assuming that successive bowls are independent, determine the probability that Alex first hits the wicket on her third attempt.
  2. Determine the probability that Alex hits the wicket for the fourth time on her tenth attempt.
Edexcel FS1 2023 June Q7
  1. Each time a spinner is spun, the probability that it lands on red is 0.2
    1. Find the probability that the spinner lands on red
      1. for the 1st time on the 4th spin
      2. for the 3rd time on the 8th spin
      3. exactly 4 times during 10 spins
    Each time the spinner is spun, the probability that it lands on yellow is 0.4
    In a game with this spinner, a player must choose one of two events
    \(R\) is the event that the spinner lands on red for the \(\mathbf { 1 s t }\) time in at most 4 spins
    \(Y\) is the event that the spinner lands on yellow for the 3rd time in at most 7 spins
  2. Showing your calculations clearly, determine which of these events has the greater probability.