Probabilities in table form with k

Probabilities already displayed in a table with expressions involving k (e.g., 3c, 4c, 5c or 4p, 5p², 1.5p) where k is found by summing the table entries and setting equal to 1.

6 questions

CAIE S1 2003 November Q8
8 A discrete random variable \(X\) has the following probability distribution.
\(x\)1234
\(\mathrm { P } ( X = x )\)\(3 c\)\(4 c\)\(5 c\)\(6 c\)
  1. Find the value of the constant \(c\).
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
  3. Find \(\mathrm { P } ( X > \mathrm { E } ( X ) )\).
CAIE S1 2010 November Q1
1 The discrete random variable \(X\) takes the values 1, 4, 5, 7 and 9 only. The probability distribution of \(X\) is shown in the table.
\(x\)14579
\(\mathrm { P } ( X = x )\)\(4 p\)\(5 p ^ { 2 }\)\(1.5 p\)\(2.5 p\)\(1.5 p\)
Find \(p\).
Edexcel S1 2011 January Q6
  1. The discrete random variable \(X\) has the probability distribution
\(x\)1234
\(\mathrm { P } ( X = x )\)\(k\)\(2 k\)\(3 k\)\(4 k\)
  1. Show that \(k = 0.1\) Find
  2. \(\mathrm { E } ( X )\)
  3. \(\mathrm { E } \left( X ^ { 2 } \right)\)
  4. \(\operatorname { Var } ( 2 - 5 X )\) Two independent observations \(X _ { 1 }\) and \(X _ { 2 }\) are made of \(X\).
  5. Show that \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } = 4 \right) = 0.1\)
  6. Complete the probability distribution table for \(X _ { 1 } + X _ { 2 }\)
    \(y\)2345678
    \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } = y \right)\)0.010.040.100.250.24
  7. Find \(\mathrm { P } \left( 1.5 < X _ { 1 } + X _ { 2 } \leqslant 3.5 \right)\)
Edexcel S1 2012 January Q3
3. The discrete random variable \(X\) can take only the values \(2,3,4\) or 6 . For these values the probability distribution function is given by
\(x\)2346
\(\mathrm { P } ( X = x )\)\(\frac { 5 } { 21 }\)\(\frac { 2 k } { 21 }\)\(\frac { 7 } { 21 }\)\(\frac { k } { 21 }\)
where \(k\) is a positive integer.
  1. Show that \(k = 3\) Find
  2. \(\mathrm { F } ( 3 )\)
  3. \(\mathrm { E } ( X )\)
  4. \(\mathrm { E } \left( X ^ { 2 } \right)\)
  5. \(\operatorname { Var } ( 7 X - 5 )\)
AQA Paper 3 2018 June Q11
11 The table below shows the probability distribution for a discrete random variable \(X\).
\(\boldsymbol { x }\)12345
\(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)\(k\)\(2 k\)\(4 k\)\(2 k\)\(k\)
Find the value of \(k\). Circle your answer.
\(\frac { 1 } { 2 }\)\(\frac { 1 } { 4 }\)\(\frac { 1 } { 10 }\)1
SPS SPS SM Mechanics 2021 January Q6
6. The discrete random variable \(D\) has the following probability distribution
\(d\)1020304050
\(\mathrm { P } ( D = d )\)\(\frac { k } { 10 }\)\(\frac { k } { 20 }\)\(\frac { k } { 30 }\)\(\frac { k } { 40 }\)\(\frac { k } { 50 }\)
where \(k\) is a constant.
  1. Show that the value of \(k\) is \(\frac { 600 } { 137 }\) The random variables \(D _ { 1 }\) and \(D _ { 2 }\) are independent and each have the same distribution as \(D\).
  2. Find P \(\left( D _ { 1 } + D _ { 2 } = 80 \right)\) Give your answer to 3 significant figures. A single observation of \(D\) is made.
    The value obtained, \(d\), is the common difference of an arithmetic sequence.
    The first 4 terms of this arithmetic sequence are the angles, measured in degrees, of quadrilateral \(Q\)
  3. Find the exact probability that the smallest angle of \(Q\) is more than \(50 ^ { \circ }\)