5. Manon has two biased spinners, one red and one green.
The random variable \(R\) represents the score when the red spinner is spun.
The random variable \(G\) represents the score when the green spinner is spun.
The probability distributions for \(R\) and \(G\) are given below.
| \(r\) | 2 | 3 |
| \(\mathrm { P } ( R = r )\) | \(\frac { 1 } { 4 }\) | \(\frac { 3 } { 4 }\) |
| \(g\) | 1 | 4 |
| \(\mathrm { P } ( G = g )\) | \(\frac { 2 } { 3 }\) | \(\frac { 1 } { 3 }\) |
Manon spins each spinner once and adds the two scores.
- Find the probability that
- the sum of the two scores is 7
- the sum of the two scores is less than 4
The random variable \(X = m R + n G\) where \(m\) and \(n\) are integers.
$$\mathrm { P } ( X = 20 ) = \frac { 1 } { 6 } \quad \text { and } \quad \mathrm { P } ( X = 50 ) = \frac { 1 } { 4 }$$
- Find the value of \(m\) and the value of \(n\)