Independent events test

3 A fair eight-sided die has faces marked \(1,2,3,4,5,6,7,8\). The score when the die is thrown is the number on the face the die lands on. The die is thrown twice.

• Event \(R\) is 'one of the scores is exactly 3 greater than the other score'.
• Event \(S\) is 'the product of the scores is more than 19'.
  1. Find the probability of \(R\).
  2. Find the probability of \(S\).
  3. Determine whether events \(R\) and \(S\) are independent. Justify your answer.

7 Bag \(A\) contains 4 balls numbered 2, 4, 5, 8. Bag \(B\) contains 5 balls numbered 1, 3, 6, 8, 8. Bag \(C\) contains 7 balls numbered \(2,7,8,8,8,8,9\). One ball is selected at random from each bag.

• Event \(X\) is 'exactly two of the selected balls have the same number'.
• Event \(Y\) is 'the ball selected from bag \(A\) has number 4'.
  1. Find \(\mathrm { P } ( X )\).
  2. Find \(\mathrm { P } ( X \cap Y )\) and hence determine whether or not events \(X\) and \(Y\) are independent.
  3. Find the probability that two balls are numbered 2, given that exactly two of the selected balls have the same number.
     

7 You are given that \(P ( A ) = 0.6 , P ( B ) = 0.5\) and \(P ( A \cup B ) ^ { \prime } = 0.2\).

1. Find \(\mathrm { P } ( \mathrm { A } \cap \mathrm { B } )\).
2. Find \(\mathrm { P } ( \mathrm { A } \mid \mathrm { B } )\).
3. State, with a reason, whether \(A\) and \(B\) are independent.

Events \(A\) and \(B\) are such that \(\text{P}(A) = 0.3\), \(\text{P}(B) = 0.8\) and \(\text{P}(A \text{ and } B) = 0.4\). State, giving a reason in each case, whether events \(A\) and \(B\) are

1. independent, [2]
2. mutually exclusive. [2]