Calculate probabilities using independence

1 Independent random variables \(X\) and \(Y\) have distributions \(\mathrm { B } ( 7 , p )\) and \(\mathrm { B } ( 8 , p )\) respectively.

1. Explain why \(X + Y \sim \mathrm {~B} ( 15 , p )\).
2. Find \(\mathrm { P } ( X = 2 \mid X + Y = 5 )\).

1. (a) State in words the relationship between two events \(R\) and \(S\) when \(\mathrm { P } ( R \cap S ) = 0\)

The events \(A\) and \(B\) are independent with \(\mathrm { P } ( A ) = \frac { 1 } { 4 }\) and \(\mathrm { P } ( A \cup B ) = \frac { 2 } { 3 }\) Find
(b) \(\mathrm { P } ( B )\)
(c) \(\mathrm { P } \left( A ^ { \prime } \cap B \right)\)
(d) \(\mathrm { P } \left( B ^ { \prime } \mid A \right)\)

3. The events \(A\) and \(B\) are independent such that \(\mathrm { P } ( A ) = 0.25\) and \(\mathrm { P } ( B ) = 0.30\). Find

1. \(\mathrm { P } ( A \cap B )\),
2. \(\mathrm { P } ( A \cup B )\),
3. \(\mathrm { P } \left( A B ^ { \prime } \right)\).

4. Explain what you understand by

1. a sample space,
2. an event. Two events \(A\) and \(B\) are independent, such that \(\mathrm { P } ( A ) = \frac { 1 } { 3 }\) and \(\mathrm { P } ( B ) = \frac { 1 } { 4 }\).
   Find
3. \(\mathrm { P } ( A \cap B )\),
4. \(\mathrm { P } ( A B )\),
5. \(\mathrm { P } ( A \cup B )\).

2. Events \(A\) and \(B\) are independent. Given also that $$\mathrm { P } ( A ) = \frac { 3 } { 4 } \quad \text { and } \quad \mathrm { P } \left( A \cap B ^ { \prime } \right) = \frac { 1 } { 4 }$$ Find

1. \(\mathrm { P } ( A \cap B )\),
2. \(\mathrm { P } ( B )\),
3. \(\mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \right)\).