Combined event algebra

Questions requiring manipulation of probability formulas involving unions, intersections, and complements to find unknown probabilities, often solving simultaneous equations.

7 questions

Edexcel S1 2018 January Q2
2. (a) Shade the region representing the event \(A \cup B ^ { \prime }\) on the Venn diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{01259350-0119-4500-a81b-bfa1b4234559-06_355_563_306_694} The two events \(C\) and \(D\) are mutually exclusive.
Given that \(\mathrm { P } ( C ) = \frac { 1 } { 5 }\) and \(\mathrm { P } ( D ) = \frac { 3 } { 10 }\) find
(b) (i) \(\quad \mathrm { P } ( C \cup D )\)
(ii) \(\mathrm { P } ( C \mid D )\) The two events \(F\) and \(G\) are independent.
Given that \(\mathrm { P } ( F ) = \frac { 1 } { 6 }\) and \(\mathrm { P } ( F \cup G ) = \frac { 3 } { 8 }\) find
(c) (i) \(\mathrm { P } ( G )\)
(ii) \(\mathrm { P } \left( F \mid G ^ { \prime } \right)\)
Edexcel S1 2024 January Q6
  1. The events \(A\) and \(B\) satisfy
$$\mathrm { P } ( A ) = x \quad \mathrm { P } ( B ) = y \quad \mathrm { P } ( A \cup B ) = 0.65 \quad \mathrm { P } ( B \mid A ) = 0.3$$
  1. Show that $$14 x + 20 y = 13$$ The events \(B\) and \(C\) are mutually exclusive such that $$\mathrm { P } ( B \cup C ) = 0.85 \quad \mathrm { P } ( C ) = \frac { 1 } { 2 } x + y$$
    1. Find a second equation in \(x\) and \(y\)
    2. Hence find the value of \(x\) and the value of \(y\)
  3. Determine whether or not \(A\) and \(B\) are statistically independent. You must show your working clearly.
Edexcel S1 2004 January Q4
4. \(\quad\) The events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = \frac { 2 } { 5 } , \mathrm { P } ( B ) = \frac { 1 } { 2 }\) and \(\mathrm { P } \left( A \quad B ^ { \prime } \right) = \frac { 4 } { 5 }\).
  1. Find
    1. \(\mathrm { P } \left( A \cap B ^ { \prime } \right)\),
    2. \(\mathrm { P } ( A \cap B )\),
    3. \(\mathrm { P } ( A \cup B )\),
    4. \(\mathrm { P } \left( \begin{array} { l l } A & B \end{array} \right)\).
  2. State, with a reason, whether or \(\operatorname { not } A\) and \(B\) are
    1. mutually exclusive,
    2. independent.
Edexcel S1 2016 June Q6
6. Three events \(A , B\) and \(C\) are such that $$\mathrm { P } ( A ) = \frac { 2 } { 5 } \quad \mathrm { P } ( C ) = \frac { 1 } { 2 } \quad \mathrm { P } ( A \cup B ) = \frac { 5 } { 8 }$$ Given that \(A\) and \(C\) are mutually exclusive find
  1. \(\mathrm { P } ( A \cup C )\) Given that \(A\) and \(B\) are independent
  2. show that \(\mathrm { P } ( B ) = \frac { 3 } { 8 }\)
  3. Find \(\mathrm { P } ( A \mid B )\) Given that \(\mathrm { P } \left( C ^ { \prime } \cap B ^ { \prime } \right) = 0.3\)
  4. draw a Venn diagram to represent the events \(A , B\) and \(C\)
Edexcel S1 2018 Specimen Q6
6. Three events \(A , B\) and \(C\) are such that $$\mathrm { P } ( A ) = \frac { 2 } { 5 } \quad \mathrm { P } ( C ) = \frac { 1 } { 2 } \quad \mathrm { P } ( A \cup B ) = \frac { 5 } { 8 }$$ Given that \(A\) and \(C\) are mutually exclusive find
  1. \(\mathrm { P } ( A \cup C )\) Given that \(A\) and \(B\) are independent
  2. show that \(\mathrm { P } ( B ) = \frac { 3 } { 8 }\)
  3. Find \(\mathrm { P } ( A \mid B )\) Given that \(\mathrm { P } \left( C ^ { \prime } \cap B ^ { \prime } \right) = 0.3\)
  4. draw a Venn diagram to represent the events \(A , B\) and \(C\)
    \includegraphics[max width=\textwidth, alt={}, center]{b7500cc1-caa6-4767-bb2e-e3d70474e805-21_2260_53_312_33} \(\_\_\_\_\) VAYV SIHI NI JIIIM ION OC
    VJYV SIHI NI JIIIM ION OC
    VJYV SIHI NI JLIYM ION OC
Edexcel S1 Q4
4. The events \(A\) and \(B\) are such that $$\mathrm { P } ( A ) = 0.5 , \mathrm { P } ( B ) = 0.42 \text { and } \mathrm { P } ( A \cup B ) = 0.76$$ Find
  1. \(\mathrm { P } ( A \cap B )\),
  2. \(\quad \mathrm { P } \left( A ^ { \prime } \cup B \right)\),
  3. \(\mathrm { P } \left( B \mid A ^ { \prime } \right)\).
  4. Show that events \(A\) and \(B\) are not independent.
Edexcel Paper 3 Specimen Q4
  1. Given that
$$\mathrm { P } ( A ) = 0.35 \quad \mathrm { P } ( B ) = 0.45 \quad \text { and } \quad \mathrm { P } ( A \cap B ) = 0.13$$ find
  1. \(\mathrm { P } \left( A ^ { \prime } \mid B ^ { \prime } \right)\)
  2. Explain why the events \(A\) and \(B\) are not independent. The event \(C\) has \(\mathrm { P } ( C ) = 0.20\)
    The events \(A\) and \(C\) are mutually exclusive and the events \(B\) and \(C\) are statistically independent.
  3. Draw a Venn diagram to illustrate the events \(A , B\) and \(C\), giving the probabilities for each region.
  4. Find \(\mathrm { P } \left( [ B \cup C ] ^ { \prime } \right)\)