Conditional probability with geometric

Find conditional probabilities involving geometric distributions (e.g., given that X = n, find probability of another event).

2 questions · Standard +0.8

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OCR Further Statistics Specimen Q6
7 marks Standard +0.8
6 A bag contains 3 green counters, 3 blue counters and \(w\) white counters. Counters are selected at random, one at a time, with replacement, until a white counter is drawn.
The total number of counters selected, including the white counter, is denoted by \(X\).
  1. In the case when \(w = 2\),
    1. write down the distribution of \(X\),
    2. find \(P ( 3 < X \leq 7 )\).
    3. In the case when \(\mathrm { E } ( X ) = 2\), determine the value of \(w\).
    4. In the case when \(w = 2\) and \(X = 6\), find the probability that the first five counters drawn alternate in colour.
OCR S1 2010 January Q9
7 marks Standard +0.8
\(R\) and \(S\) are independent random variables each having the distribution Geo\((p)\).
  1. Find P\((R = 1\) and \(S = 1)\) in terms of \(p\). [1]
  2. Show that P\((R = 3\) and \(S = 3) = p^2q^4\), where \(q = 1 - p\). [1]
  3. Use the formula for the sum to infinity of a geometric series to show that $$\text{P}(R = S) = \frac{p}{2-p}.$$ [5]