OCR S4 2018 June — Question 3 10 marks

Exam BoardOCR
ModuleS4 (Statistics 4)
Year2018
SessionJune
Marks10
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Mark schemeDownload PDF ↗
TopicPrinciple of Inclusion/Exclusion
TypeConditional Probability from Venn Diagrams
DifficultyStandard +0.8 Parts (i)-(iii) are routine applications of basic probability formulas (inclusion-exclusion, complement, conditional probability). Part (iv) requires setting up and solving a system of equations using the three-set inclusion-exclusion principle with multiple constraints, demanding careful algebraic manipulation and systematic reasoning beyond standard textbook exercises.
Spec2.03a Mutually exclusive and independent events2.03b Probability diagrams: tree, Venn, sample space2.03d Calculate conditional probability: from first principles
3 Events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = 0.6 , \mathrm { P } ( B ) = 0.4\) and \(\mathrm { P } ( A \cup B ) = 0.8\).
  1. Find \(\mathrm { P } ( A \cap B )\).
  2. Find \(\mathrm { P } \left( A \cap B ^ { \prime } \right)\).
  3. Find \(\mathrm { P } ( A \mid B )\). Events \(A\) and \(B\) are as above and a third event \(C\) is such that \(\mathrm { P } ( A \cup B \cup C ) = 1 , \mathrm { P } ( A \cap B \cap C ) = 0.05\), \(\mathrm { P } ( A \cap C ) = \mathrm { P } ( B \cap C )\) and \(\mathrm { P } \left( A \cap B ^ { \prime } \cap C ^ { \prime } \right) = 3 \mathrm { P } \left( A ^ { \prime } \cap B \cap C ^ { \prime } \right)\).
  4. Find \(\mathrm { P } ( C )\).
Question 3(i):
AnswerMarks Guidance
AnswerMarks Guidance
\(0.6 + 0.4 - 0.8\)M1
\(= 0.2\)A1, [2]
Question 3(ii):
AnswerMarks Guidance
AnswerMarks Guidance
\(0.4\)B1, [1]
Question 3(iii):
AnswerMarks Guidance
AnswerMarks Guidance
\(0.2 / 0.4\)M1
\(0.5\)A1, [2]
Question 3(iv):
AnswerMarks Guidance
AnswerMarks Guidance
\(P(A' \cap B' \cap C) = 0.2\) soiB1
Attempt to set up 2 simultaneous equations e.g. \(P(A' \cap B \cap C') = x\), \(P(A \cap C) = P(B \cap C) = y\)M1
\(3x + y = 0.4\)A1 Both
\(x + y = 0.2\)A1 Both
\(x = 0.1\), \(y = 0.1\)A1
\(P(C) = 0.45\)A1, [5] 
## Question 3(i):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $0.6 + 0.4 - 0.8$ | M1 | |
| $= 0.2$ | A1, [2] | |

## Question 3(ii):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $0.4$ | B1, [1] | |

## Question 3(iii):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $0.2 / 0.4$ | M1 | |
| $0.5$ | A1, [2] | |

## Question 3(iv):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(A' \cap B' \cap C) = 0.2$ soi | B1 | |
| Attempt to set up 2 simultaneous equations e.g. $P(A' \cap B \cap C') = x$, $P(A \cap C) = P(B \cap C) = y$ | M1 | |
| $3x + y = 0.4$ | A1 | Both |
| $x + y = 0.2$ | A1 | Both |
| $x = 0.1$, $y = 0.1$ | A1 | |
| $P(C) = 0.45$ | A1, [5] | |

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3 Events $A$ and $B$ are such that $\mathrm { P } ( A ) = 0.6 , \mathrm { P } ( B ) = 0.4$ and $\mathrm { P } ( A \cup B ) = 0.8$.\\
(i) Find $\mathrm { P } ( A \cap B )$.\\
(ii) Find $\mathrm { P } \left( A \cap B ^ { \prime } \right)$.\\
(iii) Find $\mathrm { P } ( A \mid B )$.

Events $A$ and $B$ are as above and a third event $C$ is such that $\mathrm { P } ( A \cup B \cup C ) = 1 , \mathrm { P } ( A \cap B \cap C ) = 0.05$, $\mathrm { P } ( A \cap C ) = \mathrm { P } ( B \cap C )$ and $\mathrm { P } \left( A \cap B ^ { \prime } \cap C ^ { \prime } \right) = 3 \mathrm { P } \left( A ^ { \prime } \cap B \cap C ^ { \prime } \right)$.\\
(iv) Find $\mathrm { P } ( C )$.

\hfill \mbox{\textit{OCR S4 2018 Q3 [10]}}
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