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.

3 questions · Moderate -0.8

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CAIE S1 2012 June Q3
7 marks Moderate -0.8
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,
    1. the sum of the scores is 4,
    2. the product of the scores is less than 8 .
CAIE S1 2017 June Q4
6 marks Standard +0.3
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 \(r\).
AQA AS Paper 2 2020 June Q17
3 marks Easy -1.8
A game consists of spinning a circular wheel divided into numbered sectors as shown below. \includegraphics{figure_17} On each spin the score, \(X\), is the value shown in the sector that the arrow points to when the spinner stops. The probability of the arrow pointing at a sector is proportional to the angle subtended at the centre by that sector.
  1. Show that \(P(X = 1) = \frac{5}{18}\) [1 mark]
  2. Complete the probability distribution for \(X\) in the table below.
    \(x\)1
    \(P(X = x)\)\(\frac{5}{18}\)
    [2 marks]