Compound event with two dice/coins

Model a geometric distribution where success is defined by a compound event (e.g., pair of heads, sum of two dice equals a value, pair of sixes).

6 questions · Moderate -0.4

5.02f Geometric distribution: conditions5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1)
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CAIE S1 2023 June Q1
4 marks Moderate -0.3
1 Two fair coins are thrown at the same time repeatedly until a pair of heads is obtained. The number of throws taken is denoted by the random variable \(X\).
  1. State the value of \(\mathrm { E } ( X )\).
  2. Find the probability that exactly 5 throws are required to obtain a pair of heads.
  3. Find the probability that fewer than 7 throws are required to obtain a pair of heads.
CAIE S1 2020 November Q2
5 marks Standard +0.3
2 An ordinary fair die is thrown until a 6 is obtained.
  1. Find the probability that obtaining a 6 takes more than 8 throws.
    Two ordinary fair dice are thrown together until a pair of 6s is obtained. The number of throws taken is denoted by the random variable \(X\).
  2. Find the expected value of \(X\).
  3. Find the probability that obtaining a pair of 6s takes either 10 or 11 throws.
CAIE S1 2021 November Q1
4 marks Moderate -0.3
1 Two fair coins are thrown at the same time. The random variable \(X\) is the number of throws of the two coins required to obtain two tails at the same time.
  1. Find the probability that two tails are obtained for the first time on the 7th throw.
  2. Find the probability that it takes more than 9 throws to obtain two tails for the first time.
CAIE S1 2020 June Q1
5 marks Moderate -0.8
1 The score when two fair six-sided dice are thrown is the sum of the two numbers on the upper faces.
  1. Show that the probability that the score is 4 is \(\frac { 1 } { 12 }\).
    The two dice are thrown repeatedly until a score of 4 is obtained. The number of throws taken is denoted by the random variable \(X\).
  2. Find the mean of \(X\).
  3. Find the probability that a score of 4 is first obtained on the 6th throw.
  4. Find \(\mathrm { P } ( X < 8 )\).
CAIE FP2 2014 June Q6
5 marks Moderate -0.8
6 A pair of coins is thrown repeatedly until a pair of heads is obtained. The number of throws taken is denoted by the random variable \(X\). State the expected value of \(X\). Find the probability that
  1. exactly 4 throws are required to obtain a pair of heads,
  2. fewer than 6 throws are required to obtain a pair of heads.
CAIE FP2 2017 November Q6
6 marks Moderate -0.3
6 A pair of fair dice is thrown repeatedly until a pair of sixes is obtained. The number of throws taken is denoted by the random variable \(X\).
  1. Find the mean value of \(X\).
  2. Find the probability that exactly 12 throws are required to obtain a pair of sixes.
  3. Find the probability that more than 12 throws are required to obtain a pair of sixes.