| Exam Board | OCR |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2014 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Principle of Inclusion/Exclusion |
| Type | Two-Way Table to Probability |
| Difficulty | Moderate -0.3 This is a straightforward S1 statistics question involving basic two-way table manipulation and simple probability calculations. Part (i) requires solving a simple equation (5/8 of females are children), part (ii) uses inclusion-exclusion with clearly defined sets, and part (iii) involves standard 'choose 2' probability calculations. All techniques are routine for S1 with no conceptual challenges or novel problem-solving required, making it slightly easier than average. |
| Spec | 2.03a Mutually exclusive and independent events2.03b Probability diagrams: tree, Venn, sample space |
| \cline { 2 - 3 } \multicolumn{1}{c|}{} | Male | Female |
| Adults | 78 | 45 |
| Children | 52 | \(n\) |
| \cline { 2 - 3 } \multicolumn{1}{c|}{} | Male | Female |
| Adults | 6 | 4 |
| Children | 5 | 10 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{n}{n+45} = \frac{5}{8}\) or \(n:45 = 5:3\) or \(\frac{3}{8}:45 = \frac{5}{8}:n\) | M1 | \(\frac{3F}{8}=45\) & \(n=\frac{5}{8}\times F\); \(45\times\frac{8}{3}-45\); \(45\times\frac{8}{3}\times\frac{5}{8}\) — correct first step involving \(n\) or complete correct method for finding \(n\) |
| \(n = 75\) | A1 | |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{45+"75"+52}{45+"75"+52+78}\) alone oe | M1 | \(1 - \frac{78}{45+"75"+52+78}\) oe or \(\frac{"250"-78}{"250"}\) oe. Completely correct method — \(\frac{45+"75"}{" 250"} + \frac{52+"75"}{" 250"} - \frac{"75"}{" 250"}\) or \(0.48 + 0.508 - 0.48\times0.508\) |
| \(= \frac{86}{125}\) or \(\frac{172}{250}\) or \(0.688\) (3 sf) oe | A1ft | ft their integer answer to (i). eg if their (i) is 28, ans 0.616 or \(\frac{125}{203}\) M1A1ft |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{10}{25}\times\frac{6}{24}\) or \(\frac{6}{25}\times\frac{10}{24}\) seen (or \(\frac{2}{5}\times\frac{1}{4}\) or \(\frac{6}{25}\times\frac{5}{12}\)) oe | M1 | or \(\frac{10}{25}\times\frac{6}{25} + \frac{6}{25}\times\frac{10}{25}\) or \(\frac{10}{25}\times\frac{6}{25}\times 2\) oe or \(\frac{^{10}C_1\times ^6C_1}{^{25}C_2}\) oe or \(\frac{10\times6}{300}\) oe — ie allow M1 if '\(2\times\)' is omitted OR if 25 instead of 24, but not both errors. allow M1 for correct num or denom |
| \((\frac{10}{25}\times\frac{6}{24} + \frac{6}{25}\times\frac{10}{24}\) or \(\frac{10}{25}\times\frac{6}{24}\times 2)\) | ||
| \(= \frac{1}{5}\) | A1 | NB long methods may be correct, eg \((\frac{14}{25}\times\frac{10}{14})\times(\frac{11}{24}\times\frac{6}{11})\) same as \(\frac{10}{25}\times\frac{6}{24}\) |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| FA + MC or FC + MA | ||
| Either \(\frac{4}{25}\times\frac{5}{24}\times 2\) or \(\frac{10}{25}\times\frac{6}{24}\times 2\) NB ft their (iiia) | M1 | Allow \(\frac{10}{25}\times\frac{6}{25}\times 2\) or \(\frac{4}{25}\times\frac{5}{25}\times 2\) or \(\frac{10}{25}\times\frac{6}{24} + \frac{4}{25}\times\frac{5}{24}\) or \(\frac{10}{25}\times\frac{6}{25} + \frac{4}{25}\times\frac{5}{25}\) — ie allow 25 instead of 24 AND allow one case with \(\times 2\) or both cases without \(\times 2\); ie allow 25 and one of these two errors cf scheme for (iii)(a) |
| \((\frac{4}{25}\times\frac{5}{24}\times 2 + \frac{10}{25}\times\frac{6}{24}\times 2 = \frac{1}{5} + \frac{1}{15})\) | \(\frac{^{10}C_1\times ^6C_1}{^{25}C_2} + \frac{^4C_1\times ^5C_1}{^{25}C_2}\) oe or \(\frac{60+20}{300}\) oe — allow M1 if one of these fracts correct. NB \(^{25}C_2\) in denom NOT M1, cf (iii)(a) | |
| \(= \frac{4}{15}\) or \(0.267\) (3 sf) | A1 | cao |
| [2] | NB see note on long methods in 7(iiia) |
# Question 7:
## Part 7(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{n}{n+45} = \frac{5}{8}$ or $n:45 = 5:3$ or $\frac{3}{8}:45 = \frac{5}{8}:n$ | M1 | $\frac{3F}{8}=45$ & $n=\frac{5}{8}\times F$; $45\times\frac{8}{3}-45$; $45\times\frac{8}{3}\times\frac{5}{8}$ — correct first step involving $n$ or complete correct method for finding $n$ |
| $n = 75$ | A1 | |
| **[2]** | | |
## Part 7(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{45+"75"+52}{45+"75"+52+78}$ alone oe | M1 | $1 - \frac{78}{45+"75"+52+78}$ oe or $\frac{"250"-78}{"250"}$ oe. Completely correct method — $\frac{45+"75"}{" 250"} + \frac{52+"75"}{" 250"} - \frac{"75"}{" 250"}$ or $0.48 + 0.508 - 0.48\times0.508$ |
| $= \frac{86}{125}$ or $\frac{172}{250}$ or $0.688$ (3 sf) oe | A1ft | ft their integer answer to (i). eg if their (i) is 28, ans 0.616 or $\frac{125}{203}$ M1A1ft |
| **[2]** | | |
## Part 7(iii)(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{10}{25}\times\frac{6}{24}$ or $\frac{6}{25}\times\frac{10}{24}$ seen (or $\frac{2}{5}\times\frac{1}{4}$ or $\frac{6}{25}\times\frac{5}{12}$) oe | M1 | or $\frac{10}{25}\times\frac{6}{25} + \frac{6}{25}\times\frac{10}{25}$ or $\frac{10}{25}\times\frac{6}{25}\times 2$ oe or $\frac{^{10}C_1\times ^6C_1}{^{25}C_2}$ oe or $\frac{10\times6}{300}$ oe — ie allow M1 if '$2\times$' is omitted **OR** if 25 instead of 24, but not both errors. allow M1 for correct num or denom |
| $(\frac{10}{25}\times\frac{6}{24} + \frac{6}{25}\times\frac{10}{24}$ or $\frac{10}{25}\times\frac{6}{24}\times 2)$ | | |
| $= \frac{1}{5}$ | A1 | NB long methods may be correct, eg $(\frac{14}{25}\times\frac{10}{14})\times(\frac{11}{24}\times\frac{6}{11})$ same as $\frac{10}{25}\times\frac{6}{24}$ |
| **[2]** | | |
## Part 7(iii)(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| FA + MC or FC + MA | | |
| Either $\frac{4}{25}\times\frac{5}{24}\times 2$ or $\frac{10}{25}\times\frac{6}{24}\times 2$ NB ft their (iiia) | M1 | Allow $\frac{10}{25}\times\frac{6}{25}\times 2$ or $\frac{4}{25}\times\frac{5}{25}\times 2$ or $\frac{10}{25}\times\frac{6}{24} + \frac{4}{25}\times\frac{5}{24}$ or $\frac{10}{25}\times\frac{6}{25} + \frac{4}{25}\times\frac{5}{25}$ — ie allow 25 instead of 24 AND allow one case with $\times 2$ or both cases without $\times 2$; ie allow 25 and one of these two errors cf scheme for (iii)(a) |
| $(\frac{4}{25}\times\frac{5}{24}\times 2 + \frac{10}{25}\times\frac{6}{24}\times 2 = \frac{1}{5} + \frac{1}{15})$ | | $\frac{^{10}C_1\times ^6C_1}{^{25}C_2} + \frac{^4C_1\times ^5C_1}{^{25}C_2}$ oe or $\frac{60+20}{300}$ oe — allow M1 if one of these fracts correct. NB $^{25}C_2$ in denom NOT M1, cf (iii)(a) |
| $= \frac{4}{15}$ or $0.267$ (3 sf) | A1 | cao |
| **[2]** | | NB see note on long methods in 7(iiia) |
---7 The table shows the numbers of members of a swimming club in certain categories.
\begin{center}
\begin{tabular}{ | l | c | c | }
\cline { 2 - 3 }
\multicolumn{1}{c|}{} & Male & Female \\
\hline
Adults & 78 & 45 \\
\hline
Children & 52 & $n$ \\
\hline
\end{tabular}
\end{center}
It is given that $\frac { 5 } { 8 }$ of the female members are children.\\
(i) Find the value of $n$.\\
(ii) Find the probability that a member chosen at random is either female or a child (or both).
The table below shows the corresponding numbers for an athletics club.
\begin{center}
\begin{tabular}{ | l | c | c | }
\cline { 2 - 3 }
\multicolumn{1}{c|}{} & Male & Female \\
\hline
Adults & 6 & 4 \\
\hline
Children & 5 & 10 \\
\hline
\end{tabular}
\end{center}
(iii) Two members of the athletics club are chosen at random for a photograph.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that one of these members is a female child and the other is an adult male.
\item Find the probability that exactly one of these members is female and exactly one is a child.
\end{enumerate}
\hfill \mbox{\textit{OCR S1 2014 Q7 [8]}}