Probability distribution finding parameters

Questions giving a probability distribution table with unknown parameter(s) and requiring their value(s) using the fact that probabilities sum to 1.

2 questions

CAIE S1 2012 June Q3
3 A spinner has 5 sides, numbered 1, 2, 3, 4 and 5 . When the spinner is spun, the score is the number of the side on which it lands. The score is denoted by the random variable \(X\), which has the probability distribution shown in the table.
\(x\)12345
\(\mathrm { P } ( X = x )\)0.30.15\(3 p\)\(2 p\)0.05
  1. Find the value of \(p\). A second spinner has 3 sides, numbered 1, 2 and 3. The score when this spinner is spun is denoted by the random variable \(Y\). It is given that \(\mathrm { P } ( Y = 1 ) = 0.3 , \mathrm { P } ( Y = 2 ) = 0.5\) and \(\mathrm { P } ( Y = 3 ) = 0.2\).
  2. Find the probability that, when both spinners are spun together,
    (a) the sum of the scores is 4,
    (b) the product of the scores is less than 8 .
CAIE S1 2017 June Q4
4 Two identical biased triangular spinners with sides marked 1,2 and 3 are spun. For each spinner, the probabilities of landing on the sides marked 1,2 and 3 are \(p , q\) and \(r\) respectively. The score is the sum of the numbers on the sides on which the spinners land. You are given that \(\mathrm { P } (\) score is \(6 ) = \frac { 1 } { 36 }\) and \(\mathrm { P } (\) score is \(5 ) = \frac { 1 } { 9 }\). Find the values of \(p , q\) and \(r\).