| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2015 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Principle of Inclusion/Exclusion |
| Type | Standard Survey to Venn Diagram |
| Difficulty | Moderate -0.8 This is a standard S1 inclusion-exclusion problem with straightforward Venn diagram construction and basic probability calculations. All steps follow routine procedures: filling in the diagram from the center outward, then reading off values for probability questions. The independence check at the end is also a standard textbook exercise requiring only P(B∩C) = P(B)×P(C). |
| Spec | 2.03a Mutually exclusive and independent events2.03b Probability diagrams: tree, Venn, sample space2.03c Conditional probability: using diagrams/tables2.03d Calculate conditional probability: from first principles |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| 3 intersecting circles with 3 in the centre | B1 | B1 for 3 intersecting circles with 3 in centre; allow probs. or integers |
| Values 2, 4, 8 correct (at least one correct subtraction) | M1, A1 | M1 for some correct subtraction e.g. at least one of 2, 4, 8 or for \(B\): \(20-\text{their}(2+3+4)\) etc |
| Values 11, 13, 17 correct | A1 | Must be in compatible regions with 2, 4, 8 if no labels |
| Correct labels and 22 and box | B1 | Do not treat "blank" as 0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{13}{80}\) or \(0.1625\) | B1ft | ft their diagram |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{28+30-11}{80}\) or \(\frac{2+3+4+8+13+17}{80}\) or \(1-\frac{(11+22)}{80}=\frac{47}{80}\) or \(0.5875\) | M1 A1 | M1 for correct expression (or ft their diagram) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{"17+8+13"}{" 47"}\) or \(\frac{\frac{"38"}{80}}{\frac{"47"}{80}}\) or \(1-\frac{"2+3+4"}{" 47"}=\frac{38}{47}\) (condone awrt 0.809) | M1 A1cao | M1 for denominator of 47 or ft their numerator from (c) and numerator of 38 or their \((17+8+13)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P(B\ | C)=\frac{7}{28}\), \(P(B)=\frac{20}{80}\) | M1 |
| \(P(C\ | B)=\frac{7}{20}\), \(P(C)=\frac{28}{80}\) | |
| \(P(B\cap C)=\frac{7}{80}\), \(P(B)=\frac{20}{80}P(C)=\frac{28}{80}\) | M1 | M1 for use of correct test with \(B\) and \(C\); must see product attempted for \(P(B\cap C)\) test |
| \(P(B\ | C)=P(B)\), \(P(C\ | B)=P(C)\) these may be implied by correct conclusion; \(P(B\cap C)=P(B)\times P(C)\) this approach requires the product to be seen |
| So, they are independent | A1 | A1 for correct test with all probabilities correct and correct concluding statement; NB M0M1A0 possible but A1 requires both Ms |
# Question 3:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| 3 intersecting circles with 3 in the centre | B1 | B1 for 3 intersecting circles with 3 in centre; allow probs. or integers |
| Values 2, 4, 8 correct (at least one correct subtraction) | M1, A1 | M1 for some correct subtraction e.g. at least one of 2, 4, 8 or for $B$: $20-\text{their}(2+3+4)$ etc |
| Values 11, 13, 17 correct | A1 | Must be in compatible regions with 2, 4, 8 if no labels |
| Correct labels and 22 and box | B1 | Do not treat "blank" as 0 |
## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{13}{80}$ or $0.1625$ | B1ft | ft their diagram |
## Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{28+30-11}{80}$ or $\frac{2+3+4+8+13+17}{80}$ or $1-\frac{(11+22)}{80}=\frac{47}{80}$ or $0.5875$ | M1 A1 | M1 for correct expression (or ft their diagram) |
## Part (d)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{"17+8+13"}{" 47"}$ or $\frac{\frac{"38"}{80}}{\frac{"47"}{80}}$ or $1-\frac{"2+3+4"}{" 47"}=\frac{38}{47}$ (condone awrt 0.809) | M1 A1cao | M1 for denominator of 47 or ft their numerator from (c) and numerator of 38 or their $(17+8+13)$ |
## Part (e)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P(B\|C)=\frac{7}{28}$, $P(B)=\frac{20}{80}$ | M1 | M1 for stating at least the required probabilities and labelled for correct test |
| $P(C\|B)=\frac{7}{20}$, $P(C)=\frac{28}{80}$ | | |
| $P(B\cap C)=\frac{7}{80}$, $P(B)=\frac{20}{80}P(C)=\frac{28}{80}$ | M1 | M1 for use of correct test with $B$ and $C$; must see product attempted for $P(B\cap C)$ test |
| $P(B\|C)=P(B)$, $P(C\|B)=P(C)$ these may be implied by correct conclusion; $P(B\cap C)=P(B)\times P(C)$ this approach requires the product to be seen | | |
| So, they are independent | A1 | A1 for correct test with all probabilities correct and correct concluding statement; NB M0M1A0 possible but A1 requires both Ms |
---\begin{enumerate}
\item A college has 80 students in Year 12.
\end{enumerate}
20 students study Biology\\
28 students study Chemistry\\
30 students study Physics\\
7 students study both Biology and Chemistry\\
11 students study both Chemistry and Physics\\
5 students study both Physics and Biology\\
3 students study all 3 of these subjects\\
(a) Draw a Venn diagram to represent this information.
A Year 12 student at the college is selected at random.\\
(b) Find the probability that the student studies Chemistry but not Biology or Physics.\\
(c) Find the probability that the student studies Chemistry or Physics or both.
Given that the student studies Chemistry or Physics or both,\\
(d) find the probability that the student does not study Biology.\\
(e) Determine whether studying Biology and studying Chemistry are statistically independent.\\
\hfill \mbox{\textit{Edexcel S1 2015 Q3 [13]}}