12 Rob has two six-sided dice, each with sides numbered 1, 2, 3, 4, 5, 6.
One dice is fair. The other dice is biased, with probabilities as shown in the table.
| Biased die |
| \(y\) | 1 | 2 | 3 | 4 | 5 | 6 |
| \(\mathrm { P } ( Y = y )\) | 0.3 | 0.25 | 0.2 | 0.14 | 0.1 | 0.01 |
Rob throws each dice once and notes the two scores, \(X\) on the fair dice and \(Y\) on the biased dice. He then calculates the value of the variable \(S\) which is defined as follows.
- If \(X \leqslant 3\), then \(S = X + 2 Y\).
- If \(X > 3\), then \(S = X + Y\).
- (a) Draw up a sample space diagram showing all the possible outcomes and the corresponding values of \(S\).
(b) On your diagram, circle the four cells where the value \(S = 10\) occurs. - Explain the mistake in the following calculation.
$$\mathrm { P } ( S = 10 ) = \frac { \text { Number of outcomes giving } S = 10 } { \text { Total number of outcomes } } = \frac { 4 } { 36 } = \frac { 1 } { 9 } .$$
Find the correct value of \(\mathrm { P } ( S = 10 )\).Given that \(S = 10\), find the probability that the score on one of the dice is 4 .The events " \(X = 1\) or 2 " and " \(S = n\) " are mutually exclusive. Given that \(\mathrm { P } ( S = n ) \neq 0\), find the value of \(n\).