| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2006 |
| 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 straightforward application of inclusion-exclusion with all values provided directly. Part (a) requires systematic filling of a Venn diagram (routine bookwork), while parts (b)-(e) involve reading values from the diagram and basic probability calculations. No problem-solving insight needed, just careful arithmetic. |
| Spec | 2.03a Mutually exclusive and independent events2.03b Probability diagrams: tree, Venn, sample space2.03c Conditional probability: using diagrams/tables |
| Answer | Marks |
|---|---|
| 3 closed curves that intersect; Subtract at either stage | M1, M1 |
| 9, 7, 5 or 10, 6, 12 or 41 & box | A1, A1, A1 |
| Answer | Marks |
|---|---|
| \(P(G, LH, D) = \frac{10}{100} = \frac{1}{10}\) | B1 |
| Answer | Marks |
|---|---|
| \(P(G, LH, D) = \frac{41}{100}\) | B1 |
| Answer | Marks |
|---|---|
| \(P(\text{Only two attributes}) = \frac{9+7+5}{100} = \frac{21}{100}\) | M1A1 |
| Answer | Marks |
|---|---|
| \(P(G \mid LH \& DH) = \frac{P(G \& LH \& DH)}{P(LH \& DH)} = \frac{\frac{10}{100}}{\frac{15}{100}} = \frac{10}{15} = \frac{2}{3}\) | M1A1 |
| awrt 0.667 | |
| N.B. Assumption of independence M0 |
**Part (a):**
| 3 closed curves that intersect; Subtract at either stage | M1, M1 |
| 9, 7, 5 or 10, 6, 12 or 41 & box | A1, A1, A1 |
**Part (b):**
| $P(G, LH, D) = \frac{10}{100} = \frac{1}{10}$ | B1 |
**Part (c):**
| $P(G, LH, D) = \frac{41}{100}$ | B1 |
**Part (d):**
| $P(\text{Only two attributes}) = \frac{9+7+5}{100} = \frac{21}{100}$ | M1A1 |
**Part (e):**
| $P(G \mid LH \& DH) = \frac{P(G \& LH \& DH)}{P(LH \& DH)} = \frac{\frac{10}{100}}{\frac{15}{100}} = \frac{10}{15} = \frac{2}{3}$ | M1A1 |
| awrt 0.667 | |
| N.B. Assumption of independence M0 | |
**Total 13 marks**\begin{enumerate}
\item A group of 100 people produced the following information relating to three attributes. The attributes were wearing glasses, being left handed and having dark hair.\\
Glasses were worn by 36 people, 28 were left handed and 36 had dark hair. There were 17 who wore glasses and were left handed, 19 who wore glasses and had dark hair and 15 who were left handed and had dark hair. Only 10 people wore glasses, were left handed and had dark hair.\\
(a) Represent these data on a Venn diagram.
\end{enumerate}
A person was selected at random from this group.\\
Find the probability that this person\\
(b) wore glasses but was not left handed and did not have dark hair,\\
(c) did not wear glasses, was not left handed and did not have dark hair,\\
(d) had only two of the attributes,\\
(e) wore glasses given that they were left handed and had dark hair.
\hfill \mbox{\textit{Edexcel S1 2006 Q6 [13]}}