| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2005 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conditional Probability |
| Type | Bayes with sampling without replacement |
| Difficulty | Standard +0.3 This is a straightforward application of the law of total probability and Bayes' theorem with sampling without replacement. Part (i) requires calculating P(both toffees) by conditioning on which box is chosen, using hypergeometric probabilities. Part (ii) applies Bayes' theorem directly using the result from (i). The calculations are routine with clear structure and no conceptual surprises—slightly easier than average due to small numbers and standard methodology. |
| Spec | 2.03b Probability diagrams: tree, Venn, sample space2.03c Conditional probability: using diagrams/tables2.03d Calculate conditional probability: from first principles |
| Answer | Marks | Guidance |
|---|---|---|
| For one correct 3-factor term | B1 | |
| For summing three 3-factor or 2-factor probs | M1 | |
| For correct answer | A1 | 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| For choosing only their \(P(A \text{ T T})\) in numer or denom | M1 | |
| For dividing by their (i) or what they think is \(P(T,T)\) | M1 | |
| For correct answer using either 2 or 3-term probs; Constant prob B0M1A0M1M1A0 max | A1 | 3 marks |
**(i)** $P(T, T) = \frac{1}{3} \cdot \frac{6}{10} \cdot \frac{5}{9} + \frac{1}{3} \cdot \frac{5}{8} \cdot \frac{4}{7} + \frac{1}{3} \cdot \frac{3}{10} \cdot \frac{2}{9} = 53/210$ (0.252)
| For one correct 3-factor term | B1
| For summing three 3-factor or 2-factor probs | M1
| For correct answer | A1 | 3 marks
**(ii)** $P(A \text{ TT}) = 0.111/0.252 = 70/159$ (0.440)
| For choosing only their $P(A \text{ T T})$ in numer or denom | M1
| For dividing by their (i) or what they think is $P(T,T)$ | M1
| For correct answer using either 2 or 3-term probs; Constant prob B0M1A0M1M1A0 max | A1 | 3 marks2 Boxes of sweets contain toffees and chocolates. Box $A$ contains 6 toffees and 4 chocolates, box $B$ contains 5 toffees and 3 chocolates, and box $C$ contains 3 toffees and 7 chocolates. One of the boxes is chosen at random and two sweets are taken out, one after the other, and eaten.\\
(i) Find the probability that they are both toffees.\\
(ii) Given that they are both toffees, find the probability that they both came from box $A$.
\hfill \mbox{\textit{CAIE S1 2005 Q2 [6]}}