| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tree Diagrams |
| Type | Conditional probability from tree |
| Difficulty | Moderate -0.8 This is a standard S1 tree diagram question with straightforward conditional probability calculations. Part (a) is routine setup, (b) uses the law of total probability with given percentages, while (c) and (d) apply Bayes' theorem in a textbook context with no conceptual challenges—all calculations are mechanical once the tree is drawn. |
| 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 |
|---|---|---|
| Probability tree diagram with correct structure and labels | B3 | |
| \((0.65 \times 0.7) + (0.35 \times 0.45) = 0.6125\) (\(\frac{49}{80}\)) | M2 A1 | |
| \(P(1^{\text{st}} \text{ serve in } \mid \text{ won}) = \frac{P(1^{\text{st}} \text{ serve in } \cap \text{ won})}{P(\text{won})}\) | M1 | |
| \(= \frac{0.65 \times 0.7}{0.6125} = 0.743\) (3sf) (\(\frac{26}{35}\)) | M1 A1 | |
| \(P(1^{\text{st}} \text{ serve not in } \mid \text{ lost}) = \frac{P(1^{\text{st}} \text{ serve not in } \cap \text{ lost})}{P(\text{lost})}\) | M1 | |
| \(= \frac{0.35 \times 0.55}{1 - 0.6125} = 0.497\) (3sf) (\(\frac{77}{155}\)) | M2 A2 | (14) |
| Probability tree diagram with correct structure and labels | B3 | |
| $(0.65 \times 0.7) + (0.35 \times 0.45) = 0.6125$ ($\frac{49}{80}$) | M2 A1 | |
| $P(1^{\text{st}} \text{ serve in } \mid \text{ won}) = \frac{P(1^{\text{st}} \text{ serve in } \cap \text{ won})}{P(\text{won})}$ | M1 | |
| $= \frac{0.65 \times 0.7}{0.6125} = 0.743$ (3sf) ($\frac{26}{35}$) | M1 A1 | |
| $P(1^{\text{st}} \text{ serve not in } \mid \text{ lost}) = \frac{P(1^{\text{st}} \text{ serve not in } \cap \text{ lost})}{P(\text{lost})}$ | M1 | |
| $= \frac{0.35 \times 0.55}{1 - 0.6125} = 0.497$ (3sf) ($\frac{77}{155}$) | M2 A2 | (14) |6. Serving against his regular opponent, a tennis player has a $65 \%$ chance of getting his first serve in. If his first serve is in he then has a $70 \%$ chance of winning the point but if his first serve is not in, he only has a $45 \%$ chance of winning the point.
\begin{enumerate}[label=(\alph*)]
\item Represent this information on a tree diagram.
For a point on which this player served to his regular opponent, find the probability that
\item he won the point,
\item his first serve went in given that he won the point,
\item his first serve didn't go in given that he lost the point.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 Q6 [14]}}