Finding Unknown Probabilities in Venn Diagrams

A question is this type if and only if it gives a Venn diagram with algebraic unknowns (like p, q, r) and constraints, requiring you to solve for these unknown probability values.

5 questions · Standard +0.2

2.03a Mutually exclusive and independent events2.03c Conditional probability: using diagrams/tables
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OCR MEI S1 2005 January Q3
3 marks Moderate -0.3
3 The Venn diagram illustrates the occurrence of two events \(A\) and \(B\). \includegraphics[max width=\textwidth, alt={}, center]{b35b2b3b-0d26-4a35-b4d2-110bf270d5dc-2_513_826_1713_658} You are given that \(\mathrm { P } ( A \cap B ) = 0.3\) and that the probability that neither \(A\) nor \(B\) occurs is 0.1 . You are also given that \(\mathrm { P } ( A ) = 2 \mathrm { P } ( B )\). Find \(\mathrm { P } ( B )\).
OCR MEI S1 Q5
3 marks Standard +0.3
5 The Venn diagram illustrates the occurrence of two events \(A\) and \(B\). \includegraphics[max width=\textwidth, alt={}, center]{64f25a40-d3bf-4212-b92e-655f980c702b-5_480_771_452_655} You are given that \(\mathrm { P } ( A \cap B ) = 0.3\) and that the probability that neither \(A\) nor \(B\) occurs is 0.1 . You are also given that \(\mathrm { P } ( A ) = 2 \mathrm { P } ( B )\). Find \(\mathrm { P } ( B )\).
Edexcel S1 2019 January Q1
9 marks Standard +0.3
  1. The Venn diagram shows the probability of a randomly selected student from a school being in the sets \(L , B\) and \(C\), where \(L\) represents the event that the student has instrumental music lessons \(B\) represents the event that the student plays in the school band \(C\) represents the event that the student sings in the school choir \(p , q , r\) and \(s\) are probabilities. \includegraphics[max width=\textwidth, alt={}, center]{d3f4450d-60eb-49b6-be1b-d2fcfad0451f-02_504_750_735_598}
    1. Select a pair of mutually exclusive events from \(L , B\) and \(C\).
    Given that \(\mathrm { P } ( L ) = 0.4 , \mathrm { P } ( B ) = 0.13 , \mathrm { P } ( C ) = 0.3\) and the events \(L\) and \(C\) are independent,
  2. find the value of \(p\),
  3. find the value of \(q\), the value of \(r\) and the value of \(s\). A student is selected at random from those who play in the school band or sing in the school choir.
  4. Find the exact probability that this student has instrumental music lessons.
Edexcel S1 2023 January Q4
13 marks Standard +0.3
    1. In the Venn diagram below, \(A\) and \(B\) represent events and \(p , q , r\) and \(s\) are probabilities. \includegraphics[max width=\textwidth, alt={}, center]{c316fa29-dedc-4890-bd82-31eb0bb819f9-12_400_789_347_639}
$$\mathrm { P } ( A ) = \frac { 7 } { 25 } \quad \mathrm { P } ( B ) = \frac { 1 } { 5 } \quad \mathrm { P } \left[ \left( A \cap B ^ { \prime } \right) \cup \left( A ^ { \prime } \cap B \right) \right] = \frac { 8 } { 25 }$$
  1. Use algebra to show that \(2 p + 2 q + 2 r = \frac { 4 } { 5 }\)
  2. Find the value of \(p\), the value of \(q\), the value of \(r\) and the value of \(s\) (ii) Two events, \(C\) and \(D\), are such that $$\mathrm { P } ( C ) = \frac { x } { x + 5 } \quad \mathrm { P } ( D ) = \frac { 5 } { x }$$ where \(x\) is a positive constant.
    By considering \(\mathrm { P } ( C ) + \mathrm { P } ( D )\) show that \(C\) and \(D\) cannot be mutually exclusive.
OCR MEI S1 Q3
8 marks Standard +0.3
3 The Venn diagram illustrates the occurrence of two events \(A\) and \(B\). \includegraphics[max width=\textwidth, alt={}, center]{1ad9c390-b42f-47d8-86c5-f73a42d97721-02_513_826_1713_658} You are given that \(\mathrm { P } ( A \cap B ) = 0.3\) and that the probability that neither \(A\) nor \(B\) occurs is 0.1 . You are also given that \(\mathrm { P } ( A ) = 2 \mathrm { P } ( B )\). Find \(\mathrm { P } ( B )\).