Mutually exclusive events

4 Two fair dice are thrown.

1. Event \(A\) is 'the scores differ by 3 or more'. Find the probability of event \(A\).
2. Event \(B\) is 'the product of the scores is greater than 8 '. Find the probability of event \(B\).
3. State with a reason whether events \(A\) and \(B\) are mutually exclusive.

1. The Venn diagram, where \(p , q\) and \(r\) are probabilities, shows the events \(A , B , C\) and \(D\) and associated probabilities.

\begin{figure}[h] \end{figure}

1. State any pair of mutually exclusive events from \(A\), \(B\), \(C\) and \(D\) The events \(B\) and \(C\) are independent.
2. Find the value of \(p\)
3. Find the greatest possible value of \(\mathrm { P } \left( A \mid B ^ { \prime } \right)\) Given that \(\mathrm { P } \left( B \mid A ^ { \prime } \right) = 0.5\)
4. find the value of \(q\) and the value of \(r\)
5. Find \(\mathrm { P } \left( [ A \cup B ] ^ { \prime } \cap C \right)\)
6. Use set notation to write an expression for the event with probability \(p\)

4. \begin{figure}[h] \end{figure} Figure 1 shows how 25 people travelled to work.
Their travel to work is represented by the events $$\begin{array} { l l } B & \text { bicycle } \\ T & \text { train } \\ W & \text { walk } \end{array}$$

1. Write down 2 of these events that are mutually exclusive. Give a reason for your answer.
2. Determine whether or not \(B\) and \(T\) are independent events. One person is chosen at random.
   Find the probability that this person
3. walks to work,
4. travels to work by bicycle and train.
5. Given that this person travels to work by bicycle, find the probability that they will also take the train.

A and B are mutually exclusive events. Which one of the following statements must be correct? Tick (\(\checkmark\)) one box. [1 mark] \(P(A \cup B) = P(A) \times P(B)\) \(P(A \cup B) = P(A) - P(B)\) \(P(A \cap B) = 0\) \(P(A \cap B) = 1\)