Conditional probability with algebraic expressions

A question is this type if and only if it requires manipulating algebraic expressions involving probabilities (often with variables like x, p, q) to find relationships or specific values.

8 questions

OCR S4 2007 June Q1
1 For the events \(A\) and \(B , \mathrm { P } ( A ) = 0.3 , \mathrm { P } ( B ) = 0.6\) and \(\mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \right) = c\), where \(c \neq 0\).
  1. Find \(\mathrm { P } ( A \cap B )\) in terms of \(c\).
  2. Find \(\mathrm { P } ( B \mid A )\) and deduce that \(0.1 \leqslant c \leqslant 0.4\).
OCR S4 2011 June Q3
3 For the events \(A\) and \(B , \mathrm { P } ( A ) = \mathrm { P } ( B ) = \frac { 3 } { 4 }\) and \(\mathrm { P } \left( A \mid B ^ { \prime } \right) = \frac { 1 } { 2 }\).
  1. Find \(\mathrm { P } ( A \cap B )\). For a third event \(C , \mathrm { P } ( C ) = \frac { 1 } { 4 }\) and \(C\) is independent of the event \(A \cap B\).
  2. Find \(\mathrm { P } ( A \cap B \cap C )\).
  3. Given that \(\mathrm { P } ( C \mid A ) = \lambda\) and \(\mathrm { P } ( B \mid C ) = 3 \lambda\), and that no event occurs outside \(A \cup B \cup C\), find the value of \(\lambda\).
OCR S4 2012 June Q8
8 Events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = 0.3\) and \(\mathrm { P } ( A \mid B ) = 0.6\).
  1. Show that \(\mathrm { P } ( B ) \leqslant 0.5\).
  2. Given also that \(\mathrm { P } ( A \cup B ) = x\), find \(\mathrm { P } ( B )\) in terms of \(x\).
Edexcel S1 2015 January Q4
4. Events \(A\) and \(B\) are shown in the Venn diagram below
where \(x , y , 0.10\) and 0.32 are probabilities.
\includegraphics[max width=\textwidth, alt={}, center]{c58f3e88-2dbc-40d6-a966-a5765a7c67ba-08_467_798_408_575}
  1. Find an expression in terms of \(x\) for
    1. \(\mathrm { P } ( A )\)
    2. \(\mathrm { P } ( B \mid A )\)
  2. Find an expression in terms of \(x\) and \(y\) for \(\mathrm { P } ( A \cup B )\) Given that \(\mathrm { P } ( A ) = 2 \mathrm { P } ( B )\)
  3. find the value of \(x\) and the value of \(y\)
Edexcel S1 2022 October Q6
  1. The Venn diagram shows the events \(A , B , C\) and \(D\), where \(p , q , r\) and \(s\) are probabilities.
    \includegraphics[max width=\textwidth, alt={}, center]{1fda59cb-059e-4850-810f-cc3e69bc058e-20_504_826_296_621}
    1. Write down the value of
      1. \(\mathrm { P } ( A )\)
      2. \(\mathrm { P } ( A \mid B )\)
      3. \(\mathrm { P } ( A \mid C )\)
    Given that \(\mathrm { P } \left( B ^ { \prime } \cap D ^ { \prime } \right) = \frac { 7 } { 10 }\) and \(\mathrm { P } ( C \mid D ) = \frac { 3 } { 5 }\)
  2. find the exact value of \(q\) and the exact value of \(r\) Given also that \(\mathrm { P } \left( B \cup C ^ { \prime } \right) = \frac { 5 } { 8 }\)
  3. find the exact value of \(s\)
Edexcel S1 2013 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4cf4f2d7-d912-4b65-a666-caa37009661a-11_606_1131_210_411} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The Venn diagram in Figure 1 shows three events \(A , B\) and \(C\) and the probabilities associated with each region of \(B\). The constants \(p , q\) and \(r\) each represent probabilities associated with the three separate regions outside \(B\). The events \(A\) and \(B\) are independent.
  1. Find the value of \(p\). Given that \(\mathrm { P } ( B \mid C ) = \frac { 5 } { 11 }\)
  2. find the value of \(q\) and the value of \(r\).
  3. Find \(\mathrm { P } ( A \cup C \mid B )\).
Edexcel S1 2017 June Q3
  1. The Venn diagram shows three events \(A , B\) and \(C\), where \(p , q , r , s\) and \(t\) are probabilities.
    \includegraphics[max width=\textwidth, alt={}, center]{319667e7-3f8b-4a33-8fc5-ef72154d1421-10_647_972_306_488}
    (b) Find the value of \(r\).
    (c) Hence write down the value of \(s\) and the value of \(t\).
    (d) State, giving a reason, whether or not the events \(A\) and \(B\) are independent.
    (e) Find \(\mathrm { P } ( B \mid A \cup C )\).
    \(\mathrm { P } ( A ) = 0.5 , \mathrm { P } ( B ) = 0.6\) and \(\mathrm { P } ( C ) = 0.25\) and the events \(B\) and \(C\) are independent.
    (a) Find the value of \(p\) and the value of \(q\).
SPS SPS SM Statistics 2021 May Q5
5. Two events C and D are such that \(P ( C \mid D ) = 3 \times P ( C )\) where \(P ( C ) \neq 0\).
  1. Explain whether or not events C and D could be independent events. Given also that $$P ( C \cap D ) = \frac { 1 } { 2 } \times P ( C ) \text { and } P \left( C ^ { \prime } \cap D ^ { \prime } \right) = \frac { 7 } { 10 }$$
  2. find \(P ( C )\), showing your working clearly.