Test independence using definition

1 Two ordinary fair dice, one red and the other blue, are thrown.
Event \(A\) is 'the score on the red die is divisible by 3 '.
Event \(B\) is 'the sum of the two scores is at least 9 '.

1. Find \(\mathrm { P } ( A \cap B )\).
2. Hence determine whether or not the events \(A\) and \(B\) are independent.

4 Tim throws a fair die twice and notes the number on each throw.

1. Tim calculates his final score as follows. If the number on the second throw is a 5 he multiplies the two numbers together, and if the number on the second throw is not a 5 he adds the two numbers together. Find the probability that his final score is
   1. 12,
   2. 5 .
   3. Events \(A , B , C\) are defined as follows. \(A\) : the number on the second throw is 5 \(B\) : the sum of the numbers is 6 \(C\) : the product of the numbers is even
      By calculation find which pairs, if any, of the events \(A , B\) and \(C\) are independent.

2 Ashfaq throws two fair dice and notes the numbers obtained. \(R\) is the event 'The product of the two numbers is 12 '. \(T\) is the event 'One of the numbers is odd and one of the numbers is even'. By finding appropriate probabilities, determine whether events \(R\) and \(T\) are independent.

3 A fair six-sided die is thrown twice and the scores are noted. Event \(X\) is defined as 'The total of the two scores is 4'. Event \(Y\) is defined as 'The first score is 2 or 5'. Are events \(X\) and \(Y\) independent? Justify your answer.

1 Two ordinary fair dice are thrown and the numbers obtained are noted. Event \(S\) is 'The sum of the numbers is even'. Event \(T\) is 'The sum of the numbers is either less than 6 or a multiple of 4 or both'. Showing your working, determine whether the events \(S\) and \(T\) are independent.

2 Each day Anna drives to work.

• \(R\) is the event that it is raining.
• \(L\) is the event that Anna arrives at work late.

You are given that \(\mathrm { P } ( R ) = 0.36 , \mathrm { P } ( L ) = 0.25\) and \(\mathrm { P } ( R \cap L ) = 0.2\).

1. Determine whether the events \(R\) and \(L\) are independent.
2. Draw a Venn diagram showing the events \(R\) and \(L\). Fill in the probability corresponding to each of the four regions of your diagram.
3. Find \(\mathrm { P } ( L \mid R )\). State what this probability represents.

A contractor bids for two building projects. He estimates that the probability of winning the first project is 0.5, the probability of winning the second is 0.3 and the probability of winning both projects is 0.2.

1. Find the probability that he does not win either project. [3]
2. Find the probability that he wins exactly one project. [2]
3. Given that he does not win the first project, find the probability that he wins the second. [2]
4. By calculation, determine whether or not winning the first contract and winning the second contract are independent events. [3]

\(A\), \(B\) and \(C\) are three events such that \(\text{P}(A) = x\), \(\text{P}(B) = y\) and \(\text{P}(C) = x + y\). It is known that \(\text{P}(A \cup B) = 0.6\) and \(\text{P}(B \mid A) = 0.2\).

1. Show that \(4x + 5y = 3\). [2 marks]

It is also known that \(B\) and \(C\) are mutually exclusive and that \(\text{P}(B \cup C) = 0.9\)

1. Obtain another equation in \(x\) and \(y\) and hence find the values of \(x\) and \(y\). [4 marks]
2. Deduce whether or not \(A\) and \(B\) are independent events. [2 marks]

In a survey, a large number of young people are asked about their exercise habits. One of these people is selected at random. • \(G\) is the event that this person goes to the gym. • \(R\) is the event that this person goes running. You are given that P(G) = 0.24, P(R) = 0.13 and P(G ∩ R) = 0.06.

1. Draw a Venn diagram, showing the events \(G\) and \(R\), and fill in the probability corresponding to each of the four regions of your diagram. [3]
2. Determine whether the events \(G\) and \(R\) are independent. [2]
3. Find P(R | G). [3]

The table shows all the possible products of the scores on two fair four-sided dice.

1. Find the probability that the product of the two scores is less than 10. [1]
2. Show that the events 'the score on the first die is even' and 'the product of the scores on the two dice is less than 10' are not independent. [3]

In a recent survey, a large number of working people were asked whether they worked full-time or part-time, with part-time being defined as less than 25 hours per week. One of the respondents is selected at random. • \(W\) is the event that this person works part-time. • \(F\) is the event that this person is female. You are given that P(\(W\)) = 0.14, P(\(F\)) = 0.41 and P(\(W \cap F\)) = 0.11.

1. Draw a Venn diagram showing the events \(W\) and \(F\), and fill in the probability corresponding to each of the four regions of your diagram. [3]
2. Determine whether the events \(W\) and \(F\) are independent. [2]
3. Find P(\(W\) | \(F\)) and explain what this probability represents. [3]

In a recent survey, a large number of working people were asked whether they worked full-time or part-time, with part-time being defined as less than 25 hours per week. One of the respondents is selected at random. • \(W\) is the event that this person works part-time. • \(F\) is the event that this person is female. You are given that \(\text{P}(W) = 0.14\), \(\text{P}(F) = 0.41\) and \(\text{P}(W \cap F) = 0.11\).

1. Draw a Venn diagram showing the events \(W\) and \(F\), and fill in the probability corresponding to each of the four regions of your diagram. [3]
2. Determine whether the events \(W\) and \(F\) are independent. [2]
3. Find \(\text{P}(W|F)\) and explain what this probability represents. [3]

The table shows all the possible products of the scores on two fair four-sided dice. \includegraphics{figure_6}

1. Find the probability that the product of the two scores is less than 10. [1]
2. Show that the events 'the score on the first die is even' and 'the product of the scores on the two dice is less than 10' are not independent. [3]

\(P(E) = 0.25\), \(P(F) = 0.4\) and \(P(E \cap F) = 0.12\)

1. Find \(P(E'|F')\) [2 marks]
2. Explain, showing your working, whether or not \(E\) and \(F\) are statistically independent. Give reasons for your answer. [2 marks]

The event \(G\) has \(P(G) = 0.15\) The events \(E\) and \(G\) are mutually exclusive and the events \(F\) and \(G\) are independent.

1. Draw a Venn diagram to illustrate the events \(E\), \(F\) and \(G\), giving the probabilities for each region. [3 marks]
2. Find \(P([F \cup G]')\) [2 marks]

\(A\) and \(B\) are two events. You are given that \(\mathrm{P}(A) = 0.6\), \(\mathrm{P}(B) = 0.5\) and \(\mathrm{P}(A \cup B) = 0.8\).

1. Find \(\mathrm{P}(A \cap B)\). [2]
2. Find \(\mathrm{P}(B | A)\). [2]
3. Explain whether the events \(A\) and \(B\) are independent or not. [1]