Independent events test

Questions asking whether events are independent by checking if P(A∩B) = P(A)×P(B), or requiring calculation given independence.

6 questions

CAIE S1 2010 June Q5
5 Two fair twelve-sided dice with sides marked \(1,2,3,4,5,6,7,8,9,10,11,12\) are thrown, and the numbers on the sides which land face down are noted. Events \(Q\) and \(R\) are defined as follows.
\(Q\) : the product of the two numbers is 24 .
\(R\) : both of the numbers are greater than 8 .
  1. Find \(\mathrm { P } ( Q )\).
  2. Find \(\mathrm { P } ( R )\).
  3. Are events \(Q\) and \(R\) exclusive? Justify your answer.
  4. Are events \(Q\) and \(R\) independent? Justify your answer.
CAIE S1 2016 March Q3
3 A fair eight-sided die has faces marked \(1,2,3,4,5,6,7,8\). The score when the die is thrown is the number on the face the die lands on. The die is thrown twice.
  • Event \(R\) is 'one of the scores is exactly 3 greater than the other score'.
  • Event \(S\) is 'the product of the scores is more than 19'.
    1. Find the probability of \(R\).
    2. Find the probability of \(S\).
    3. Determine whether events \(R\) and \(S\) are independent. Justify your answer.
CAIE S1 2020 Specimen Q7
7 Bag \(A\) ch ais \(4 \mathbf { b }\) lls \(\mathrm { m } \quad \mathbf { b }\) red \(2,4,58\) Bag \(B\) ch ais \(5 \mathbf { b }\) lls \(\mathrm { m } \quad \mathbf { b }\) red 1,3688 Bag \(C\) co ais 7 b lls m b redram a b \(l l\) is selected \(t\) rach frm eaclb \(g\)
  • Ed \(n X\) is 'ed ctlyt wo th selecteb lls \(\mathbf { h }\) th same m br'.
  • Ed n \(Y\) is 'tb b ll selected rm bag \(A \mathbf { h }\) sm br4.
    1. FidP (X).
    2. Fid ( \(X \cap Y\) ) aid \(\mathbf { n }\) ed termin wh ther or \(\mathbf { n }\) even \(\mathrm { s } X\) ad \(Y\) are id \(\mathbf { p } \mathbf { d } \quad \mathrm { h }\). [B
    3. Fid the p b b lity th t two \(\mathbf { b }\) lls are \(\mathrm { m } \quad \mathbf { b }\) red \(2 \dot { \mathrm {~g} }\) n th t ex ctly two \(\mathbf { 6 }\) th selected \(\mathbf { b }\) lls h \& th same m br.
If B e th follw ig lin dpg to cm p ete th an wer(s) to ay q stin (s), th q stin \(\mathrm { m } \quad \mathbf { b } \quad \mathrm { r } ( \mathrm { s } )\) ms tb clearlys n n
OCR MEI Paper 2 2020 November Q7
7 You are given that \(P ( A ) = 0.6 , P ( B ) = 0.5\) and \(P ( A \cup B ) ^ { \prime } = 0.2\).
  1. Find \(\mathrm { P } ( \mathrm { A } \cap \mathrm { B } )\).
  2. Find \(\mathrm { P } ( \mathrm { A } \mid \mathrm { B } )\).
  3. State, with a reason, whether \(A\) and \(B\) are independent.
Edexcel S1 Q1
  1. Given that \(\mathrm { P } ( A \cup B ) = 0.65 , \mathrm { P } ( A \cap B ) = 0.15\) and \(\mathrm { P } ( A ) = 0.3\), determine, with explanation, whether or not the events \(A\) and \(B\) are
    1. mutually exclusive,
    2. independent.
    3. (a) Give one example in each case of a quantity which could be modelled as
      1. a discrete random variable,
      2. a continuous random variable.
    4. Name one discrete distribution and one continuous distribution, stating clearly which is which.
    5. A regular tetrahedron has its faces numbered \(1,2,3\) and 4 . It is weighted so that when it is thrown, the probability of each face being in contact with the table is inversely proportional to the number on that face. This number is represented by the random variable \(X\).
    6. Show that \(\mathrm { P } ( X = 1 ) = \frac { 12 } { 25 }\) and find the probabilities of the other values of \(X\).
    7. Calculate the mean and the variance of \(X\).
    8. The random variable \(X\) is normally distributed with mean 17 . The probability that \(X\) is less than 16 is 0-3707.
    9. Calculate the standard deviation of \(X\).
    10. In 75 independent observations of \(X\), how many would you expect to be greater than 20?
    11. The students in a large Sixth Form can choose to do exactly one of Community Service, Games or Private Study on Wednesday afternoons. The probabilities that a randomly chosen student does Games and Private Study are \(\frac { 3 } { 8 }\) and \(\frac { 1 } { 5 }\) respectively. It may be assumed that the number of students is large enough for these probabilities to be treated as constant.
    12. Find the probability that a randomly chosen student does Community Service.
    13. If two students are chosen at random, find the probability that they both do the same activity.
    14. If three students are chosen at random, find the probability that exactly one of them does Games.
    Two-fifths of the students are girls, and a quarter of these girls do Private Study.
  2. Find the probability that a randomly chosen student who does Private Study is a boy. \section*{STATISTICS 1 (A)TEST PAPER 10 Page 2}
SPS SPS FM Statistics 2024 April Q6
6. For the events \(A\) and \(B\), $$\mathrm { P } \left( A \cap B ^ { \prime } \right) = 0.32 , \quad \mathrm { P } \left( A ^ { \prime } \cap B \right) = 0.11 \quad \text { and } \quad \mathrm { P } ( A \cup B ) = 0.65$$
  1. Find \(\mathrm { P } \left( A \mid B ^ { \prime } \right)\).
  2. Determine whether or not \(A\) and \(B\) are independent.