Listing outcomes and counting

2 Ivan throws three fair dice.

1. List all the possible scores on the three dice which give a total score of 5 , and hence show that the probability of Ivan obtaining a total score of 5 is \(\frac { 1 } { 36 }\).
2. Find the probability of Ivan obtaining a total score of 7.

5 A school athletics team has 10 members. The table shows which competitions each of the members can take part in. An athlete is selected at random. Events \(A , B , C , D\) are defined as follows.
A: the athlete can take part in exactly 2 competitions.
\(B\) : the athlete can take part in the 200 m .
\(C\) : the athlete can take part in the 110 m hurdles.
\(D\) : the athlete can take part in the long jump.

1. Write down the value of \(\mathrm { P } ( A \cap B )\).
2. Write down the value of \(\mathrm { P } ( C \cup D )\).
3. Which two of the four events \(A , B , C , D\) are mutually exclusive?
4. Show that events \(B\) and \(D\) are not independent.

5 Sophie and James are having a tennis competition. The winner of the competition is the first to win 2 matches in a row. If the competition has not been decided after 5 matches, then the player who has won more matches is declared the winner of the competition. For example, the following sequences are two ways in which Sophie could win the competition. (S represents a match won by Sophie; \(\mathbf { J }\) represents a match won by James.) \section*{SJSS SJSJS}

1. Explain why the sequence \(\mathbf { S S J }\) is not possible.
2. Write down the other three possible sequences in which Sophie wins the competition.
3. The probability that Sophie wins a match is 0.7 . Find the probability that she wins the competition in no more than 4 matches.

3 Sophie and James are having a tennis competition. The winner of the competition is the first to win 2 matches in a row. If the competition has not been decided after 5 matches, then the player who has won more matches is declared the winner of the competition. For example, the following sequences are two ways in which Sophie could win the competition. ( \(\mathbf { S }\) represents a match won by Sophie; \(\mathbf { J }\) represents a match won by James.) \section*{SJSS SJSJS}

1. Explain why the sequence \(\mathbf { S S J }\) is not possible.
2. Write down the other three possible sequences in which Sophie wins the competition.
3. The probability that Sophie wins a match is 0.7 . Find the probability that she wins the competition in no more than 4 matches.

1 A school athletics team has 10 members. The table shows which competitions each of the members can take part in. An athlete is selected at random. Events \(A , B , C , D\) are defined as follows.
\(A\) : the athlete can take part in exactly 2 competitions.
\(B\) : the athlete can take part in the 200 m .
\(C\) : the athlete can take part in the 110 m hurdles.
\(D\) : the athlete can take part in the long jump.

1. Write down the value of \(\mathrm { P } ( A \cap B )\).
2. Write down the value of \(\mathrm { P } ( C \cup D )\).
3. Which two of the four events \(A , B , C , D\) are mutually exclusive?
4. Show that events \(B\) and \(D\) are not independent.

15 Numbered balls are placed in bowls A, B and C In bowl A there are four balls numbered 1, 2, 3 and 7 In bowl B there are eight balls numbered \(0,0,2,3,5,6,8\) and 9
In bowl \(C\) there are nine balls numbered \(0,1,1,2,3,3,3,6\) and 7
This information is shown in the diagram below. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} A three-digit number is generated using the following method:

• a ball is selected at random from each bowl
• the first digit of the number is the ball drawn from bowl A
• the second digit of the number is the ball drawn from bowl B
• the third digit of the number is the ball drawn from bowl C

15

1. Find the probability that the number generated is even.
   15
2. Find the probability that the number generated is 703
   [2 marks]