Combined event algebra

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) \(\mathrm { P } \left( F \mid G ^ { \prime } \right)\)

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 $$\mathrm { P } ( B \cup C ) = 0.85 \quad \mathrm { P } ( C ) = \frac { 1 } { 2 } x + y$$
2. 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.

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.

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 \(C\)

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

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.

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 \(\mathrm { P } \left( [ B \cup C ] ^ { \prime } \right)\)

Three events \(A\), \(B\) and \(C\) are such that $$\mathrm{P}(A) = 0.1 \quad \mathrm{P}(B|A) = 0.3 \quad \mathrm{P}(A \cup B) = 0.25 \quad \mathrm{P}(C) = 0.5$$ Given that \(A\) and \(C\) are mutually exclusive

1. find P\((A \cup C)\) [1]
2. Show that P\((B) = 0.18\) [3]

Given also that \(B\) and \(C\) are independent,

1. draw a Venn diagram to represent the events \(A\), \(B\) and \(C\) and the probabilities associated with each region. [5]

\(A\) and \(B\) are two events such that $$P(A \cap B) = 0.1$$ $$P(A' \cap B') = 0.2$$ $$P(B) = 2P(A)$$

1. Find \(P(A)\) [4 marks]
2. Find \(P(B|A)\) [2 marks]
3. Determine if \(A\) and \(B\) are independent events. [1 mark]

The events \(A, B\) are such that \(P(A) = 0.2, P(B) = 0.3\). Determine the value of \(P(A \cup B)\) when

1. \(A,B\) are mutually exclusive, [2]
2. \(A,B\) are independent, [3]
3. \(A \subset B\). [1]

An architect bids for two construction projects. He estimates the probability of winning bid \(A\) is \(0 \cdot 6\), the probability of winning bid \(B\) is \(0 \cdot 5\) and the probability of winning both is \(0 \cdot 2\).

1. Show that the probability that he does not win either bid is \(0 \cdot 1\). [2]
2. Find the probability that he wins exactly one bid. [2]
3. Given that he does not win bid \(A\), find the probability that he wins bid \(B\). [3]