7 James has a fair coin and a fair tetrahedral die with four faces numbered 1, 2, 3, 4. He tosses the coin once and the die twice. The random variable \(X\) is defined as follows.

• If the coin shows a head then \(X\) is the sum of the scores on the two throws of the die.
• If the coin shows a tail then \(X\) is the score on the first throw of the die only.
  1. Explain why \(X = 1\) can only be obtained by throwing a tail, and show that \(\mathrm { P } ( X = 1 ) = \frac { 1 } { 8 }\).
  2. Show that \(\mathrm { P } ( X = 3 ) = \frac { 3 } { 16 }\).
  3. Copy and complete the probability distribution table for \(X\).

Event \(Q\) is 'James throws a tail'. Event \(R\) is 'the value of \(X\) is 7'.

Determine whether events \(Q\) and \(R\) are exclusive. Justify your answer.