Independent Events

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. The Venn diagram, where \(p\) and \(q\) are probabilities, shows the three events \(A , B\) and \(C\) and their associated probabilities.
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   1. Find \(\mathrm { P } ( A )\)
   The events \(B\) and \(C\) are independent.
2. Find the value of \(p\) and the value of \(q\)
3. Find \(\mathrm { P } \left( A \mid B ^ { \prime } \right)\)

1 In a group of 30 adults, 25 are right-handed and 8 wear spectacles. The number who are right-handed and do not wear spectacles is 19 .

1. Copy and complete the following table to show the number of adults in each category.

   	Wears spectacles	Does not wear spectacles	Total
   Right-handed
   Not right-handed
   Total			30

   An adult is chosen at random from the group. Event \(X\) is 'the adult chosen is right-handed'; event \(Y\) is 'the adult chosen wears spectacles'.
2. Determine whether \(X\) and \(Y\) are independent events, justifying your answer.

1 Events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = 0.3 , \mathrm { P } ( B ) = 0.8\) and \(\mathrm { P } ( A\) and \(B\) )=0.4. State, giving a reason in each case, whether events \(A\) and \(B\) are

1. independent,
2. mutually exclusive.

1 Independent random variables \(X\) and \(Y\) have distributions \(\mathrm { B } ( 7 , p )\) and \(\mathrm { B } ( 8 , p )\) respectively.

1. Explain why \(X + Y \sim \mathrm {~B} ( 15 , p )\).
2. Find \(\mathrm { P } ( X = 2 \mid X + Y = 5 )\).

2. The events \(A\) and \(B\) are independent. Given that \(\mathrm { P } ( A ) = 0.4\) and \(\mathrm { P } ( A \cap B ) = 0.12\), find

1. \(\mathrm { P } ( B )\),
2. \(\mathrm { P } ( A \cup B )\),
3. \(\mathrm { P } \left( A ^ { \prime } \cap B \right)\),
4. \(\mathrm { P } \left( A \mid B ^ { \prime } \right)\).

1 A statistics student asks people to complete a survey. The probability that a randomly chosen person agrees to complete the survey is 0.2 . Find the probability that at least one of the first three people asked agrees to complete the survey.

6. Three events \(A , B\) and \(C\) are defined in the sample space \(S\). The events \(A\) and \(B\) are mutually exclusive and \(A\) and \(C\) are independent.

1. Draw a Venn diagram to illustrate the relationships between the 3 events and the sample space. Given that \(\mathrm { P } ( A ) = 0.2 , \mathrm { P } ( B ) = 0.4\) and \(\mathrm { P } ( A \cup C ) = 0.7\), find
2. \(\mathrm { P } ( A C )\),
3. \(\mathrm { P } ( A \cup B )\),
4. \(\mathrm { P } ( C )\). END

12 You must show detailed reasoning in this question. In the summer of 2017 in England a large number of candidates sat GCSE examinations in both mathematics and English. 56\% of these candidates achieved at least level 4 in mathematics and \(80 \%\) of these candidates achieved at least level 4 in English. 14\% of these candidates did not achieve at least level 4 in either mathematics or English. Determine whether achieving level 4 or above in English and achieving level 4 or above in mathematics were independent events.

1 For the mutually exclusive events \(A\) and \(B , \mathrm { P } ( A ) = \mathrm { P } ( B ) = x\), where \(x \neq 0\).

1. Show that \(x \leqslant \frac { 1 } { 2 }\).
2. Show that \(A\) and \(B\) are not independent. The event \(C\) is independent of \(A\) and also independent of \(B\), and \(\mathrm { P } ( C ) = 2 x\).
3. Show that \(\mathrm { P } ( A \cup B \cup C ) = 4 x ( 1 - x )\).

2 Jameel has 5 plums and 3 apricots in a box. Rosa has \(x\) plums and 6 apricots in a box. One fruit is chosen at random from Jameel's box and one fruit is chosen at random from Rosa's box. The probability that both fruits chosen are plums is \(\frac { 1 } { 4 }\). Write down an equation in \(x\) and hence find \(x\). [3]

5 Marco has four boxes labelled \(K , L , M\) and \(N\). He places them in a straight line in the order \(K , L , M\), \(N\) with \(K\) on the left. Marco also has four coloured marbles: one is red, one is green, one is white and one is yellow. He places a single marble in each box, at random. Events \(A\) and \(B\) are defined as follows.
\(A\) : The white marble is in either box \(L\) or box \(M\).
\(B\) : The red marble is to the left of both the green marble and the yellow marble.
Determine whether or not events \(A\) and \(B\) are independent.