| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Session | Specimen |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tree Diagrams |
| Type | Probability distribution from tree |
| Difficulty | Moderate -0.8 This is a straightforward tree diagram question requiring basic probability calculations with independent events (0.4 success, 0.6 failure). Students must recognize the stopping condition (success or 3 failures) and compute a simple probability distribution with expected value. All techniques are standard S1 content with no novel insight required, making it easier than average. |
| Spec | 2.03b Probability diagrams: tree, Venn, sample space2.04a Discrete probability distributions5.02b Expectation and variance: discrete random variables |
| \(x\) | 0 | 1 | 2 | 3 |
| \(\mathrm { P } ( X = x )\) | 0.24 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Tree diagram with 3 pairs S/F (bank, log in, success) | 1 | M1 3 pairs S and F seen, no extra bits |
| Exactly 3 pairs, must be labelled | 1 | A1 Exactly 3 pairs, must be labelled |
| Correct diagram with probabilities 0.4 (S) and 0.6 (F) at each branch | 1 | A1 Correct diagram with all probs correct |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(0) = 0.4\) | 1 | B1 \(P(0)\) correct |
| Multiplying two or more factors of 0.4 and 0.6 | 1 | M1 Multiplying two or more factors of 0.4 and 0.6 |
| \(P(2) = 0.144\) | 1 | A1 One more correct prob |
| \(P(3) = 0.216\) | 1 | B1 One more correct prob |
| Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(E(X) = 0.24 + 2 \times 0.144 + 3 \times 0.216\) | 1 | M1 Using \(\sum p_i x_i\) |
| \(= 1.176\ (1.18)\) | 1 | A1 Correct answer |
| Total: 2 |
## Question 6(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Tree diagram with 3 pairs S/F (bank, log in, success) | 1 | **M1** 3 pairs S and F seen, no extra bits |
| Exactly 3 pairs, must be labelled | 1 | **A1** Exactly 3 pairs, must be labelled |
| Correct diagram with probabilities 0.4 (S) and 0.6 (F) at each branch | 1 | **A1** Correct diagram with all probs correct |
| **Total: 3** | | |
---
## Question 6(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(0) = 0.4$ | 1 | **B1** $P(0)$ correct |
| Multiplying two or more factors of 0.4 and 0.6 | 1 | **M1** Multiplying two or more factors of 0.4 and 0.6 |
| $P(2) = 0.144$ | 1 | **A1** One more correct prob |
| $P(3) = 0.216$ | 1 | **B1** One more correct prob |
| **Total: 4** | | |
---
## Question 6(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $E(X) = 0.24 + 2 \times 0.144 + 3 \times 0.216$ | 1 | **M1** Using $\sum p_i x_i$ |
| $= 1.176\ (1.18)$ | 1 | **A1** Correct answer |
| **Total: 2** | | |
---6 Nadia is very forgetful. Every time she logs in to her online bank she only has a $40 \%$ chance of remembering her password correctly. She is allowed 3 unsuccessful attempts on any one day and then the bank will not let her try again until the next day.\\
(i) Draw a fully labelled tree diagram to illustrate this situation.\\
(ii) Let $X$ be the number of unsuccessful attempts Nadia makes on any day that she tries to log in to her bank. Complete the following table to show the probability distribution of $X$.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
$x$ & 0 & 1 & 2 & 3 \\
\hline
$\mathrm { P } ( X = x )$ & & 0.24 & & \\
\hline
\end{tabular}
\end{center}
(iii) Calculate the expected number of unsuccessful attempts made by Nadia on any day that she tries to $\log$ in.\\
\hfill \mbox{\textit{CAIE S1 Q6 [9]}}