Geometric distribution probability

Find probabilities involving the number of trials until first success, using geometric distribution (e.g. repeated coin tosses until tails).

2 questions · Moderate -0.1

2.04d Normal approximation to binomial
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CAIE S1 2020 June Q5
9 marks Standard +0.3
5 A pair of fair coins is thrown repeatedly until a pair of tails is obtained. The random variable \(X\) denotes the number of throws required to obtain a pair of tails.
  1. Find the expected value of \(X\).
  2. Find the probability that exactly 3 throws are required to obtain a pair of tails.
  3. Find the probability that fewer than 6 throws are required to obtain a pair of tails.
    On a different occasion, a pair of fair coins is thrown 80 times.
  4. Use an approximation to find the probability that a pair of tails is obtained more than 25 times.
CAIE S1 2020 Specimen Q5
7 marks Moderate -0.5
5 A fair six-sided die, with faces marked 1, 2, 3, 4, 5, 6, is thrown 90 times.
  1. Use an approximation to find the probability that a 3 is obtained fewer than 18 times.
  2. Justify your use of the approximation in part (a).
    On another occasion, the same die is thrown repeatedly until a 3 is obtained.
  3. Find the probability that obtaining a 3 requires fewer than 7 throws.