Both independence and mutual exclusivity

5 Jasmine throws two ordinary fair 6-sided dice at the same time and notes the numbers on the uppermost faces. The events \(A\) and \(B\) are defined as follows.
\(A\) : The sum of the two numbers is less than 6 .
\(B : \quad\) The difference between the two numbers is at most 2 .

1. Determine whether or not the events \(A\) and \(B\) are independent.
2. Find \(\mathrm { P } \left( B \mid A ^ { \prime } \right)\).

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

1. independent,
2. mutually exclusive.

3 Jason throws two fair dice, each with faces numbered 1 to 6 . Event \(A\) is 'one of the numbers obtained is divisible by 3 and the other number is not divisible by 3 '. Event \(B\) is 'the product of the two numbers obtained is even'.

1. Determine whether events \(A\) and \(B\) are independent, showing your working.
2. Are events \(A\) and \(B\) mutually exclusive? Justify your answer.

3 Ellie throws two fair tetrahedral dice, each with faces numbered 1, 2, 3 and 4. She notes the numbers on the faces that the dice land on. Event \(S\) is 'the sum of the two numbers is 4 '. Event \(T\) is 'the product of the two numbers is an odd number'.

1. Determine whether events \(S\) and \(T\) are independent, showing your working.
2. Are events \(S\) and \(T\) exclusive? Justify your answer.

15 Two independent events, \(A\) and \(B\), are such that $$\begin{aligned} \mathrm { P } ( A ) & = 0.2
\mathrm { P } ( A \cup B ) & = 0.8 \end{aligned}$$ 15

1. 1. Find \(\mathrm { P } ( B )\)
      15
2. (ii) Find \(\mathrm { P } ( A \cap B )\)
   15
3. State, with a reason, whether or not the events \(A\) and \(B\) are mutually exclusive.
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