| Exam Board | OCR MEI |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2016 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Definitions |
| Type | Multiple independent trials |
| Difficulty | Moderate -0.8 This is a straightforward application of independent probability with clearly stated probabilities. Parts (i) and (ii) require simple multiplication of probabilities, while part (iii) needs basic combinatorial thinking but follows standard patterns. The question is easier than average as it involves routine probability calculations with no conceptual challenges or problem-solving insight required. |
| Spec | 2.03a Mutually exclusive and independent events2.03b Probability diagrams: tree, Venn, sample space |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(\text{Does not lose any match}) = 0.8^3 = 0.512 = \frac{64}{125}\) | B1 [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(\text{Wins all 3 or draws all 3 or loses all 3}) = 0.5^3 + 0.3^3 + 0.2^3\) | M1 | Including addition |
| \(= 0.16 = \frac{4}{25}\) | A1 [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(\text{all three outcomes occur}) = 3! \times 0.5 \times 0.3 \times 0.2\) | M1* | Allow M1 for \(k \times 0.5 \times 0.3 \times 0.2\) even if \(k=1\); Even if cubed |
| \(= 0.18\) | A1 | Not if cubed |
| Required probability \(= 1 - 0.18 - 0.16\) | *M1 dep | |
| \(= 0.66 = \frac{33}{50}\) | A1 [4] | CAO |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(WWW') + P(DDD') + P(LLL')\) | M1 | For any one product (no need for '\(3\times\)'); Even if cubed |
| \(3 \times 0.5^2 \times 0.5 + 3 \times 0.3^2 \times 0.7 + 3 \times 0.2^2 \times 0.8\) | M1 | For '\(3\times\)'; Dep on at least 1 correct term |
| \(0.375 + 0.189 + 0.096\) | M1 | For sum of three correct terms (no need for '\(3\times\)') and no incorrect terms; NB common wrong answer of 0.22 from omitting '\(3\times\)' or 0.44 from '\(2\times\)' scores M1M0M1A0; Not if cubed |
| \(0.66\) | A1 | CAO |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(WWD) + P(WWL) + P(DDW) + P(DDL) + P(LLW) + P(LLD)\) | M1 | For any one product (no need for '\(3\times\)'); Even if cubed |
| \(3\times0.5^2\times0.3+3\times0.5^2\times0.2+3\times0.3^2\times0.5+3\times0.3^2\times0.2+3\times0.2^2\times0.5+3\times0.2^2\times0.3\) | M1 | For '\(3\times\)'; Dep on at least 1 correct term |
| \(0.225 + 0.15 + 0.135 + 0.054 + 0.06 + 0.036\) | M1 | For sum of six correct terms (no need for '\(3\times\)') and no incorrect terms; Not if cubed |
| \(0.66\) | A1 | CAO |
# Question 2:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(\text{Does not lose any match}) = 0.8^3 = 0.512 = \frac{64}{125}$ | B1 [1] | |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(\text{Wins all 3 or draws all 3 or loses all 3}) = 0.5^3 + 0.3^3 + 0.2^3$ | M1 | Including addition |
| $= 0.16 = \frac{4}{25}$ | A1 [2] | |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(\text{all three outcomes occur}) = 3! \times 0.5 \times 0.3 \times 0.2$ | M1* | Allow M1 for $k \times 0.5 \times 0.3 \times 0.2$ even if $k=1$; Even if cubed |
| $= 0.18$ | A1 | Not if cubed |
| Required probability $= 1 - 0.18 - 0.16$ | *M1 dep | |
| $= 0.66 = \frac{33}{50}$ | A1 [4] | CAO |
**OR:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(WWW') + P(DDD') + P(LLL')$ | M1 | For any one product (no need for '$3\times$'); Even if cubed |
| $3 \times 0.5^2 \times 0.5 + 3 \times 0.3^2 \times 0.7 + 3 \times 0.2^2 \times 0.8$ | M1 | For '$3\times$'; Dep on at least 1 correct term |
| $0.375 + 0.189 + 0.096$ | M1 | For sum of three correct terms (no need for '$3\times$') and no incorrect terms; NB common wrong answer of 0.22 from omitting '$3\times$' or 0.44 from '$2\times$' scores M1M0M1A0; Not if cubed |
| $0.66$ | A1 | CAO |
**OR:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(WWD) + P(WWL) + P(DDW) + P(DDL) + P(LLW) + P(LLD)$ | M1 | For any one product (no need for '$3\times$'); Even if cubed |
| $3\times0.5^2\times0.3+3\times0.5^2\times0.2+3\times0.3^2\times0.5+3\times0.3^2\times0.2+3\times0.2^2\times0.5+3\times0.2^2\times0.3$ | M1 | For '$3\times$'; Dep on at least 1 correct term |
| $0.225 + 0.15 + 0.135 + 0.054 + 0.06 + 0.036$ | M1 | For sum of six correct terms (no need for '$3\times$') and no incorrect terms; Not if cubed |
| $0.66$ | A1 | CAO |
---2 In a hockey league, each team plays every other team 3 times. The probabilities that Team A wins, draws and loses to Team B are given below.
\begin{itemize}
\item $\mathrm { P } ($ Wins $) = 0.5$
\item $\mathrm { P } ($ Draws $) = 0.3$
\item $\mathrm { P } ($ Loses $) = 0.2$
\end{itemize}
The outcomes of the 3 matches are independent.\\
(i) Find the probability that Team A does not lose in any of the 3 matches.\\
(ii) Find the probability that Team A either wins all 3 matches or draws all 3 matches or loses all 3 matches.\\
(iii) Find the probability that, in the 3 matches, exactly two of the outcomes, 'Wins', 'Draws' and 'Loses' occur for Team A.
\hfill \mbox{\textit{OCR MEI S1 2016 Q2 [7]}}