| Exam Board | OCR |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2008 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Independent Events |
| Type | Prove independence relationship algebraically |
| Difficulty | Standard +0.3 This is a straightforward probability question requiring basic definitions and algebraic manipulation. Part (i) uses the fact that mutually exclusive events have P(A∪B)≤1, part (ii) applies the definition of independence to show P(A∩B)≠P(A)P(B), and part (iii) uses inclusion-exclusion with independence properties. All steps follow directly from standard definitions with minimal problem-solving insight required, making it slightly easier than average. |
| Spec | 2.03a Mutually exclusive and independent events |
| Answer | Marks | Guidance |
|---|---|---|
| Either: Obtain \(x = 0\) | B1 | ignoring errors in working |
| Form linear equation with signs of 4x and 3x different | M1 | ignoring other sign errors |
| State 4x - 5 = -3x + 5 | A1 | or equiv without brackets |
| Obtain \(\frac{10}{7}\) and no other non-zero value(s) | A1 | or exact equiv |
| Or: Obtain \(16x^2 - 40x + 25 = 9x^2 - 30x + 25\) | B1 | or equiv |
| Attempt solution of quadratic equation | M1 | at least as far as factorisation or use of formula |
| Obtain \(\frac{10}{7}\) and no other non-zero value(s) | A1 | or exact equiv |
| Obtain 0 | B1 | ignoring errors in working |
Either: Obtain $x = 0$ | B1 | ignoring errors in working
Form linear equation with signs of 4x and 3x different | M1 | ignoring other sign errors
State 4x - 5 = -3x + 5 | A1 | or equiv without brackets
Obtain $\frac{10}{7}$ and no other non-zero value(s) | A1 | or exact equiv
Or: Obtain $16x^2 - 40x + 25 = 9x^2 - 30x + 25$ | B1 | or equiv
Attempt solution of quadratic equation | M1 | at least as far as factorisation or use of formula
Obtain $\frac{10}{7}$ and no other non-zero value(s) | A1 | or exact equiv
Obtain 0 | B1 | ignoring errors in working1 For the mutually exclusive events $A$ and $B , \mathrm { P } ( A ) = \mathrm { P } ( B ) = x$, where $x \neq 0$.\\
(i) Show that $x \leqslant \frac { 1 } { 2 }$.\\
(ii) Show that $A$ and $B$ are not independent.
The event $C$ is independent of $A$ and also independent of $B$, and $\mathrm { P } ( C ) = 2 x$.\\
(iii) Show that $\mathrm { P } ( A \cup B \cup C ) = 4 x ( 1 - x )$.
\hfill \mbox{\textit{OCR S4 2008 Q1 [7]}}