One unknown from sum constraint only

Questions providing a partial probability distribution with one unknown constant (or multiple unknowns with a simple relationship) and asking to find it using only the constraint that probabilities sum to 1.

35 questions

CAIE S1 2006 November Q2
2 The discrete random variable \(X\) has the following probability distribution.
\(x\)01234
\(\mathrm { P } ( X = x )\)0.26\(q\)\(3 q\)0.050.09
  1. Find the value of \(q\).
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
CAIE S1 2007 November Q2
2 The random variable \(X\) takes the values \(- 2,0\) and 4 only. It is given that \(\mathrm { P } ( X = - 2 ) = 2 p , \mathrm { P } ( X = 0 ) = p\) and \(\mathrm { P } ( X = 4 ) = 3 p\).
  1. Find \(p\).
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
CAIE S1 2014 November Q2
2 The number of phone calls, \(X\), received per day by Sarah has the following probability distribution.
\(x\)01234\(\geqslant 5\)
\(\mathrm { P } ( X = x )\)0.240.35\(2 k\)\(k\)0.050
  1. Find the value of \(k\).
  2. Find the mode of \(X\).
  3. Find the probability that the number of phone calls received by Sarah on any particular day is more than the mean number of phone calls received per day.
CAIE S1 2018 November Q2
2 A random variable \(X\) has the probability distribution shown in the following table, where \(p\) is a constant.
\(x\)- 10124
\(\mathrm { P } ( X = x )\)\(p\)\(p\)\(2 p\)\(2 p\)0.1
  1. Find the value of \(p\).
  2. Given that \(\mathrm { E } ( X ) = 1.15\), find \(\operatorname { Var } ( X )\).
CAIE S1 2019 November Q4
4 In a probability distribution the random variable \(X\) takes the values \(- 1,0,1,2,4\). The probability distribution table for \(X\) is as follows.
\(x\)- 10124
\(\mathrm { P } ( X = x )\)\(\frac { 1 } { 4 }\)\(p\)\(p\)\(\frac { 3 } { 8 }\)\(4 p\)
  1. Find the value of \(p\).
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
  3. Given that \(X\) is greater than zero, find the probability that \(X\) is equal to 2 .
OCR S1 2005 January Q4
4 The table below shows the probability distribution of the random variable \(X\).
\(x\)- 2- 1012
\(\mathrm { P } ( X = x )\)\(\frac { 1 } { 4 }\)\(\frac { 1 } { 5 }\)\(k\)\(\frac { 2 } { 5 }\)\(\frac { 1 } { 10 }\)
  1. Find the value of the constant \(k\).
  2. Calculate the values of \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
OCR S1 2007 January Q1
1 Part of the probability distribution of a variable, \(X\), is given in the table.
\(x\)0123
\(\mathrm { P } ( X = x )\)\(\frac { 3 } { 10 }\)\(\frac { 1 } { 5 }\)\(\frac { 2 } { 5 }\)
  1. Find \(\mathrm { P } ( X = 0 )\).
  2. Find \(\mathrm { E } ( X )\).
OCR MEI S1 Q4
4 A zoologist is studying the feeding behaviour of a group of 4 gorillas. The random variable \(X\) represents the number of gorillas that are feeding at a randomly chosen moment. The probability distribution of \(X\) is shown in the table below.
\(r\)01234
\(\mathrm { P } ( X = r )\)\(p\)0.10.050.050.25
  1. Find the value of \(p\).
  2. Find the expectation and variance of \(X\).
  3. The zoologist observes the gorillas on two further occasions. Find the probability that there are at least two gorillas feeding on both occasions.
OCR S1 2012 January Q1
1 The probability distribution of a random variable \(X\) is shown in the table.
\(x\)1234
\(\mathrm { P } ( X = x )\)0.10.3\(2 p\)\(p\)
  1. Find \(p\).
  2. Find \(\mathrm { E } ( X )\).
OCR MEI S1 2009 January Q3
3 A zoologist is studying the feeding behaviour of a group of 4 gorillas. The random variable \(X\) represents the number of gorillas that are feeding at a randomly chosen moment. The probability distribution of \(X\) is shown in the table below.
\(r\)01234
\(\mathrm { P } ( X = r )\)\(p\)0.10.050.050.25
  1. Find the value of \(p\).
  2. Find the expectation and variance of \(X\).
  3. The zoologist observes the gorillas on two further occasions. Find the probability that there are at least two gorillas feeding on both occasions.
AQA Paper 3 Specimen Q11
3 marks
11 Terence owns a local shop. His shop has three checkouts, at least one of which is always staffed. A regular customer observed that the probability distribution for \(N\), the number of checkouts that are staffed at any given time during the spring, is $$\mathrm { P } ( N = n ) = \left\{ \begin{array} { c c } \frac { 3 } { 4 } \left( \frac { 1 } { 4 } \right) ^ { n - 1 } & \text { for } n = 1,2
k & \text { for } n = 3 \end{array} \right.$$ 11
  1. Find the value of \(k\).
    [0pt] [1 mark]
    11
  2. Find the probability that a customer, visiting Terence's shop during the spring, will find at least 2 checkouts staffed.
    [0pt] [2 marks]
Edexcel AS Paper 2 2024 June Q5
  1. A biased 4 -sided spinner has the numbers \(6,7,8\) and 10 on it.
The discrete random variable \(X\) represents the score when the spinner is spun once and has the following probability distribution,
\(x\)67810
\(\mathrm { P } ( X = x )\)0.50.2\(q\)\(q\)
where \(q\) is a probability.
  1. Find the value of \(q\) Karen spins the spinner repeatedly until she either gets a 7 or she has taken 4 spins.
  2. Show that the probability that Karen stops after taking her 3rd spin is 0.128 The random variable \(S\) represents the number of spins Karen takes.
  3. Find the probability distribution for \(S\) The random variable \(N\) represents the number of times Karen gets a 7
  4. Find \(\mathrm { P } ( S > N )\)
Edexcel Paper 3 2020 October Q4
  1. 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 \(\mathrm { P } \left( D _ { 1 } + D _ { 2 } = 80 \right)\) 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 \(50 ^ { \circ }\)
Edexcel Paper 3 2021 October Q6
  1. The discrete random variable \(X\) has the following probability distribution
\(x\)\(a\)\(b\)\(c\)
\(\mathrm { P } ( X = x )\)\(\log _ { 36 } a\)\(\log _ { 36 } b\)\(\log _ { 36 } c\)
where
  • \(\quad a , b\) and \(c\) are distinct integers \(( a < b < c )\)
  • all the probabilities are greater than zero
    1. Find
      1. the value of a
      2. the value of \(b\)
      3. the value of \(c\)
Show your working clearly. The independent random variables \(X _ { 1 }\) and \(X _ { 2 }\) each have the same distribution as \(X\)
  • Find \(\mathrm { P } \left( X _ { 1 } = X _ { 2 } \right)\) \section*{Question 6 continued.} \section*{Question 6 continued.}
    •
    OCR PURE Q10
    10 The probability distribution of a random variable \(X\) is given in the table.
    \(x\)0246
    \(\mathrm { P } ( X = x )\)\(\frac { 3 } { 8 }\)\(\frac { 5 } { 16 }\)\(4 p\)\(p\)
    1. Find the value of \(p\).
    2. Two values of \(X\) are chosen at random. Find the probability that the product of these values is 0 .
    OCR MEI AS Paper 2 2022 June Q1
    1 The probability distribution for the discrete random variable \(X\) is shown below.
    \(x\)12345
    \(\mathrm { P } ( \mathrm { X } = \mathrm { x } )\)0.20.15\(a\)0.270.14
    Find the value of \(a\).
    OCR MEI AS Paper 2 2023 June Q9
    9 The table shows the probability distribution for the discrete random variable \(X\).
    \(x\)12345
    \(\mathrm { P } ( \mathrm { X } = \mathrm { x } )\)0.10.3\(q\)\(2 q\)\(3 q\)
    You are given that \(q\) is a positive constant.
    1. Determine the value of \(q\).
    2. Calculate \(\mathrm { P } ( X \leqslant 4 )\). Two independent values of \(X\) are taken.
    3. Determine the probability that the sum of the two values is 3 . Fifty independent values of \(X\) are taken.
    4. Find the probability that a value of 2 occurs exactly 17 times.
    OCR MEI AS Paper 2 2021 November Q6
    6 The probability distribution for the discrete random variable \(X\) is shown below.
    \(x\)0123
    \(\mathrm { P } ( X = x )\)\(3 p ^ { 2 }\)\(0.5 p ^ { 2 } + 2 p\)\(1.5 p\)\(1.5 p ^ { 2 } + 0.5 p\)
    1. Determine the value of \(p\).
    2. Determine the modal value of \(X\).
    OCR MEI Paper 2 2019 June Q1
    1 Fig. 1 shows the probability distribution of the discrete random variable \(X\). \begin{table}[h]
    \(x\)12345
    \(\mathrm { P } ( X = x )\)0.20.1\(k\)\(2 k\)\(4 k\)
    \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{table}
    1. Find the value of \(k\).
    2. Find \(\mathrm { P } ( X \neq 4 )\).
    OCR MEI Paper 2 2023 June Q4
    4 A biased octagonal dice has faces numbered from 1 to 8 . The discrete random variable \(X\) is the score obtained when the dice is rolled once. The probability distribution of \(X\) is shown in the table below.
    \(x\)12345678
    \(\mathrm { P } ( X = x )\)\(p\)\(p\)\(p\)\(p\)\(p\)\(p\)\(p\)\(3 p\)
    1. Determine the value of \(p\).
    2. Find the probability that a score of at least 4 is obtained when the dice is rolled once. The dice is rolled 30 times.
    3. Determine the probability that a score of 8 occurs exactly twice.
    OCR MEI Paper 2 2024 June Q6
    6 The probability distribution of the discrete random variable \(X\) is shown in the table.
    \(x\)0123
    \(\mathrm { P } ( \mathrm { X } = \mathrm { x } )\)0.2\(a\)\(3 a\)0.4
    1. Calculate the value of the constant \(a\).
    2. A single value of \(X\) is chosen at random. Find the probability that the value is an odd number.
    3. Two independent values of \(X\) are chosen at random. Calculate the probability that the total of the two values is 3 .
    Edexcel S1 2001 June Q4
    4. The discrete random variable \(X\) has the probability function shown in the table below.
    \(x\)- 2- 10123
    \(\mathrm { P } ( X = x )\)0.1\(\alpha\)0.30.20.10.1
    Find
    1. \(\alpha\),
    2. \(\mathrm { P } ( - 1 < X \leq 2 )\),
    3. \(\mathrm { F } ( - 0.4 )\),
    4. \(\mathrm { E } ( 3 X + 4 )\),
    5. \(\operatorname { Var } ( 2 X + 3 )\).
    Edexcel S1 2010 June Q3
    3. The discrete random variable \(X\) has probability distribution given by
    \(x\)- 10123
    \(\mathrm { P } ( X = x )\)\(\frac { 1 } { 5 }\)\(a\)\(\frac { 1 } { 10 }\)\(a\)\(\frac { 1 } { 5 }\)
    where \(a\) is a constant.
    1. Find the value of \(a\).
    2. Write down \(\mathrm { E } ( X )\).
    3. Find \(\operatorname { Var } ( X )\). The random variable \(Y = 6 - 2 X\)
    4. Find \(\operatorname { Var } ( Y )\).
    5. Calculate \(\mathrm { P } ( X \geqslant Y )\).
    Edexcel S1 2011 June Q8
    1. A spinner is designed so that the score \(S\) is given by the following probability distribution.
    \(s\)01245
    \(\mathrm { P } ( S = s )\)\(p\)0.250.250.200.20
    1. Find the value of \(p\).
    2. Find \(\mathrm { E } ( S )\).
    3. Show that \(\mathrm { E } \left( S ^ { 2 } \right) = 9.45\)
    4. Find \(\operatorname { Var } ( S )\). Tom and Jess play a game with this spinner. The spinner is spun repeatedly and \(S\) counters are awarded on the outcome of each spin. If \(S\) is even then Tom receives the counters and if \(S\) is odd then Jess receives them. The first player to collect 10 or more counters is the winner.
    5. Find the probability that Jess wins after 2 spins.
    6. Find the probability that Tom wins after exactly 3 spins.
    7. Find the probability that Jess wins after exactly 3 spins.
    Edexcel S1 2014 June Q1
    1. The discrete random variable \(X\) has probability distribution
    \(x\)- 4- 2135
    \(\mathrm { P } ( X = x )\)0.4\(p\)0.050.15\(p\)
    1. Show that \(p = 0.2\) Find
    2. \(\mathrm { E } ( X )\)
    3. \(\mathrm { F } ( 0 )\)
    4. \(\mathrm { P } ( 3 X + 2 > 5 )\) Given that \(\operatorname { Var } ( X ) = 13.35\)
    5. find the possible values of \(a\) such that \(\operatorname { Var } ( a X + 3 ) = 53.4\)