| Exam Board | OCR |
|---|---|
| Module | PURE |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Verify probability from independent trials |
| Difficulty | Standard +0.3 This is a straightforward application of discrete probability distributions with independence. Part (a)(i) requires identifying the single scenario (1,0,0 in any order) and calculating 3 × (1/5) × (3/25)². Part (a)(ii) involves systematic enumeration of cases where two matches match and one differs. Part (b) is a basic comparison of observed vs expected mean. All techniques are standard with no novel insight required, making this slightly easier than average. |
| Spec | 2.04a Discrete probability distributions2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities |
| Number of goals | 0 | 1 | 2 | 3 | 4 |
| ||
| Probability | \(\frac { 3 } { 25 }\) | \(\frac { 1 } { 5 }\) | \(\frac { 8 } { 25 }\) | \(\frac { 7 } { 25 }\) | \(\frac { 2 } { 25 }\) | 0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\left(\frac{3}{25}\right)^2 \times \frac{1}{5}\) \((\times 3)\) | M1 | 3.1a: Correct product seen, not necessarily alone. Allow without "\(\times 3\)"; may be implied by the answer |
| \(= \frac{27}{3125}\) or \(0.00864\) ISW | A1 [2] | 1.1: NB \(\left(\frac{1}{5}\right)^3 = 0.008\) M0A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\left(\frac{3}{25}\right)^2\times\frac{22}{25}+\left(\frac{1}{5}\right)^2\times\frac{4}{5}+\left(\frac{8}{25}\right)^2\times\frac{17}{25}+\left(\frac{7}{25}\right)^2\times\frac{18}{25}+\left(\frac{2}{25}\right)^2\times\frac{23}{25}\) | M1 | 3.4: M2 for all 5 correct products |
| or \(0.0127 + 0.032 + 0.0696 + 0.0564 + 0.00589\) | M1 | 1.1: M1 for 2 correct products |
| or \(\frac{198}{15625}+\frac{4}{125}+\frac{1088}{15625}+\frac{882}{15625}+\frac{92}{15625}\) | Correct answer with no working scores M1M0A1 | |
| \(= 0.177\) (3 sf) or \(\frac{552}{3125}\) or \(\frac{2760}{15625}\) | A1 [3] | 1.1: SC. If no marks scored, but all 20 cases listed: B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Above expression | M2 | M1 for either \(P(\text{1st 2 the same})\) or \(P(\text{all 3 same})\) correct |
| \(= 0.177\) (3 sf) or \(\frac{552}{3125}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Basic scheme: Must refer to some aspect of the model that makes \(\approx 3\) goals per match unlikely. Must mention or imply \(\approx 3\) goals per match. | B1 | Guidance: oe. Or eg \(P(X > 4)\) should be more than 0, or model suggests impossible to score more than 4 goals, or model says \(P(3 \text{ or more}) = 0.36\) which is small. |
| Answer | Marks | Guidance |
|---|---|---|
| - Current model seems to underestimate probabilities of higher numbers of goals | B1 | NOT \(\approx 3\) goals per match, but \(P(X=3) = \frac{7}{25}\), too small. |
| - Or about 3 goals per match, but mean in the model is \(< 3\), or about 3 goals per match, but mean \(= 2\) or median \(= 2\) | B1 | NOT \(\approx 3\) goals per match, but \(\left(\frac{7}{25}\right)^{10}\) is tiny |
| - Or model suggests more matches \(<\) than 3 than \(> 3\) | B1 | NOT \(\approx 3\) goals per match unlikely given this model; NOT 3 is not the most likely number of goals; NOT Highest probability is 2 |
| - Or model suggests \(< 3\) goals per match | B1 | Ignore all else |
## Question 11(a)(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\left(\frac{3}{25}\right)^2 \times \frac{1}{5}$ $(\times 3)$ | M1 | 3.1a: Correct product seen, not necessarily alone. Allow without "$\times 3$"; may be implied by the answer |
| $= \frac{27}{3125}$ or $0.00864$ ISW | A1 [2] | 1.1: NB $\left(\frac{1}{5}\right)^3 = 0.008$ M0A0 |
---
## Question 11(a)(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\left(\frac{3}{25}\right)^2\times\frac{22}{25}+\left(\frac{1}{5}\right)^2\times\frac{4}{5}+\left(\frac{8}{25}\right)^2\times\frac{17}{25}+\left(\frac{7}{25}\right)^2\times\frac{18}{25}+\left(\frac{2}{25}\right)^2\times\frac{23}{25}$ | M1 | 3.4: M2 for all 5 correct products |
| or $0.0127 + 0.032 + 0.0696 + 0.0564 + 0.00589$ | M1 | 1.1: M1 for 2 correct products |
| or $\frac{198}{15625}+\frac{4}{125}+\frac{1088}{15625}+\frac{882}{15625}+\frac{92}{15625}$ | | Correct answer with no working scores M1M0A1 |
| $= 0.177$ (3 sf) or $\frac{552}{3125}$ or $\frac{2760}{15625}$ | A1 [3] | 1.1: SC. If no marks scored, but all 20 cases listed: B1 |
**Alternative method:** $P(\text{1st 2 the same}) - P(\text{all 3 same})$
$$\left(\frac{3}{25}\right)^2+\left(\frac{1}{5}\right)^2+\left(\frac{8}{25}\right)^2+\left(\frac{7}{25}\right)^2+\left(\frac{2}{25}\right)^2 - \left[\left(\frac{3}{25}\right)^3+\left(\frac{1}{5}\right)^3+\left(\frac{8}{25}\right)^3+\left(\frac{7}{25}\right)^3+\left(\frac{2}{25}\right)^3\right]$$
| Answer | Marks | Guidance |
|--------|-------|----------|
| Above expression | M2 | M1 for either $P(\text{1st 2 the same})$ or $P(\text{all 3 same})$ correct |
| $= 0.177$ (3 sf) or $\frac{552}{3125}$ | A1 | |
## Question 11:
### Part (b):
Basic scheme: Must refer to some aspect of the model that makes $\approx 3$ goals per match unlikely. Must mention or imply $\approx 3$ goals per match. | **B1** | **Guidance:** oe. Or eg $P(X > 4)$ should be more than 0, or model suggests impossible to score more than 4 goals, or model says $P(3 \text{ or more}) = 0.36$ which is small.
**Acceptable answers include:**
- Current model seems to underestimate probabilities of higher numbers of goals | B1 | NOT $\approx 3$ goals per match, but $P(X=3) = \frac{7}{25}$, too small.
- Or about 3 goals per match, but mean in the model is $< 3$, or about 3 goals per match, but mean $= 2$ or median $= 2$ | B1 | NOT $\approx 3$ goals per match, but $\left(\frac{7}{25}\right)^{10}$ is tiny
- Or model suggests more matches $<$ than 3 than $> 3$ | B1 | NOT $\approx 3$ goals per match unlikely given this model; NOT 3 is not the most likely number of goals; NOT Highest probability is 2
- Or model suggests $< 3$ goals per match | B1 | Ignore all else
**Total: [1]**11 Alex models the number of goals that a local team will score in any match as follows.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | }
\hline
Number of goals & 0 & 1 & 2 & 3 & 4 & \begin{tabular}{ c }
More \\
than 4 \\
\end{tabular} \\
\hline
Probability & $\frac { 3 } { 25 }$ & $\frac { 1 } { 5 }$ & $\frac { 8 } { 25 }$ & $\frac { 7 } { 25 }$ & $\frac { 2 } { 25 }$ & 0 \\
\hline
\end{tabular}
\end{center}
The number of goals scored in any match is independent of the number of goals scored in any other match.
\begin{enumerate}[label=(\alph*)]
\item Alex chooses 3 matches at random. Use the model to determine the probability of each of the following.
\begin{enumerate}[label=(\roman*)]
\item The team will score a total of exactly 1 goal in the 3 matches.
\item The numbers of goals scored in the first 2 of the 3 matches will be equal, but the number of goals scored in the 3rd match will be different.
During the first 10 matches this season, the team scores a total of 31 goals.
\end{enumerate}\item Without carrying out a formal test, explain briefly whether this casts doubt on the validity of Alex's model.
\section*{END OF QUESTION PAPER}
\end{enumerate}
\hfill \mbox{\textit{OCR PURE Q11 [6]}}