| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2015 |
| Session | November |
| 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 with independent trials and simple probability calculations. Students need to draw a 3-level tree (with early termination on success), calculate geometric-type probabilities using p=0.4 and q=0.6, and find an expectation. The structure is clearly guided, one probability is given, and the calculations involve only basic multiplication and addition—easier than average A-level content. |
| 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/Working | Marks | Guidance |
| Tree diagram with 3 pairs S and F | M1 | 3 pairs S (bank, log in, success oe) and F oe seen, no extra bits |
| Exactly 3 pairs, must be labelled | A1 | Exactly 3 pairs, must be labelled |
| Correct diagram with all probabilities correct | A1 [3] | Correct diagram with all probs correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P(0) = 0.4\) correct | B1 | \(P(0)\) correct |
| Multiplying two or more factors of 0.4 and 0.6 | M1 | Multiplying two or more factors of 0.4 and 0.6 |
| \(\begin{array}{c\ | c\ | c\ |
| B1 [4] | One more correct prob |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(E(X) = 0.24 + 2 \times 0.144 + 3 \times 0.216\) | M1 | Using \(\Sigma p_i x_i\) |
| \(= 1.176\ (1.18)\) | A1 [2] | Correct answer |
## Question 6 (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Tree diagram with 3 pairs S and F | M1 | 3 pairs S (bank, log in, success oe) and F oe seen, no extra bits |
| Exactly 3 pairs, must be labelled | A1 | Exactly 3 pairs, must be labelled |
| Correct diagram with all probabilities correct | A1 [3] | Correct diagram with all probs correct |
## Question 6 (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P(0) = 0.4$ correct | B1 | $P(0)$ correct |
| Multiplying two or more factors of 0.4 and 0.6 | M1 | Multiplying two or more factors of 0.4 and 0.6 |
| $\begin{array}{c\|c\|c\|c\|c} x & 0 & 1 & 2 & 3 \\ \hline \text{Prob} & 0.4 & & 0.144 & 0.216 \end{array}$ | A1 | One more correct prob |
| | B1 [4] | One more correct prob |
## Question 6 (iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $E(X) = 0.24 + 2 \times 0.144 + 3 \times 0.216$ | M1 | Using $\Sigma p_i x_i$ |
| $= 1.176\ (1.18)$ | A1 [2] | Correct answer |
---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. Copy and 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 2015 Q6 [9]}}