Find unknown probability given independence

A question is this type if and only if it states that events are independent and provides some probabilities, requiring you to solve for an unknown probability (e.g., find P(B) given P(A) and P(A ∪ B)).

3 questions

Edexcel S1 2022 June Q4
  1. The events \(H\) and \(W\) are such that
$$\mathrm { P } ( H ) = \frac { 3 } { 8 } \quad \mathrm { P } ( H \cup W ) = \frac { 3 } { 4 }$$ Given that \(H\) and \(W\) are independent,
  1. show that \(\mathrm { P } ( W ) = \frac { 3 } { 5 }\) The event \(N\) is such that $$\mathrm { P } ( N ) = \frac { 1 } { 15 } \quad \mathrm { P } ( H \cap N ) = \mathrm { P } ( N )$$
  2. Find \(\mathrm { P } \left( N ^ { \prime } \mid H \right)\) Given that \(W\) and \(N\) are mutually exclusive,
  3. draw a Venn diagram to represent the events \(H , W\) and \(N\) giving the exact probabilities of each region in the Venn diagram.
Edexcel S1 Q2
2. The events \(A\) and \(B\) are independent. Given that \(\mathrm { P } ( A ) = 0.4\) and \(\mathrm { P } ( A \cap B ) = 0.12\), find
  1. \(\mathrm { P } ( B )\),
  2. \(\mathrm { P } ( A \cup B )\),
  3. \(\mathrm { P } \left( A ^ { \prime } \cap B \right)\),
  4. \(\mathrm { P } \left( A \mid B ^ { \prime } \right)\).
Edexcel S1 Q3
3. The events \(A\) and \(B\) are such that $$\mathrm { P } ( A ) = 0.2 \text { and } \mathrm { P } ( A \cup B ) = 0.6$$ Find
  1. \(\mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \right)\),
  2. \(\quad \mathrm { P } \left( A ^ { \prime } \cap B \right)\). Given also that events \(A\) and \(B\) are independent, find
  3. \(\mathrm { P } ( B )\),
  4. \(\mathrm { P } \left( A ^ { \prime } \cup B ^ { \prime } \right)\).