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 · Moderate -0.9

2.04a Discrete probability distributions
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CAIE S1 2003 November Q8
8 marks Easy -1.3
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
3 marks Moderate -0.8
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
14 marks Moderate -0.8
  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
11 marks Moderate -0.8
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
1 marks Easy -1.8
The table below shows the probability distribution for a discrete random variable \(X\).
\(x\)12345
P(\(X = x\))\(k\)\(2k\)\(4k\)\(2k\)\(k\)
Find the value of \(k\). Circle your answer. [1 mark] \(\frac{1}{2}\) \quad \(\frac{1}{4}\) \quad \(\frac{1}{10}\) \quad \(1\)
OCR MEI Paper 2 2022 June Q11
10 marks Standard +0.3
A die in the form of a dodecahedron has its faces numbered from 1 to 12. The die is biased so that the probability that a score of 12 is achieved is different from any other score. The probability distribution of \(X\), the score on the die, is given in the table in terms of \(p\) and \(k\), where \(0 < p < 1\) and \(k\) is a positive integer.
\(x\)123456789101112
P\((X = x)\)\(p\)\(p\)\(p\)\(p\)\(p\)\(p\)\(p\)\(p\)\(p\)\(p\)\(p\)\(kp\)
Sam rolls the die 30 times, Leo rolls the die 60 times and Nina rolls the die 120 times. They each plot their scores on bar line graphs.
  1. Explain whose graph is most likely to give the best representation of the theoretical probability distribution for the score on the die. [1]
  2. Find \(p\) in terms of \(k\). [2]
  3. Determine, in terms of \(k\), the expected number of times Nina rolls a 12. [3]
  4. Given that Nina rolls a 12 on 32 occasions, calculate an estimate of the value of \(k\). [2]
Nina rolls the die a further 30 times.
  1. Use your answer to part (d) to calculate an estimate for the probability that she obtains a 12 exactly 8 times in these 30 rolls. [2]