| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2022 |
| Session | March |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Distribution |
| Type | P(X > n) or P(X ≥ n) |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question testing standard geometric distribution formulas and basic probability. Parts (a)-(b) require direct application of geometric distribution P(X=n) and P(X>n) formulas with p=3/15=1/5. Parts (c)-(e) involve routine combinations for sampling without replacement. All techniques are standard S1 content with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.03a Mutually exclusive and independent events5.02f Geometric distribution: conditions5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| [Probability of lemon \(= \frac{3}{15} = \frac{1}{5}\)] | ||
| \(\left(\frac{4}{5}\right)^6 \times \frac{1}{5} = \frac{4096}{78125}\), \(0.0524\) | B1 | \(0.0524288\) rounded to more than 3SF if final answer |
| 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\left(1-\frac{1}{5}\right)^6\) | M1 | or \(\left(\frac{4}{5}\right)^6\). FT *their* \(\frac{1}{5}\) or correct. From final answer. Condone \(\left(\frac{4}{5}\right)^5\) or \(\left(\frac{1}{5}\right)\times\left(\frac{4}{5}\right)^5 + \left(\frac{4}{5}\right)^6\) |
| \(\frac{4096}{15625}\), \(0.262\) | A1 | \(0.262144\) rounded to more than 3SF |
| Alternative method: | ||
| \([1 - P(1,2,3,4,5,[6]) =]\) \(1-\left(\frac{1}{5}+\frac{4}{5}\times\frac{1}{5}+\left(\frac{4}{5}\right)^2\times\frac{1}{5}+\left(\frac{4}{5}\right)^3\times\frac{1}{5}+\left(\frac{4}{5}\right)^4\times\frac{1}{5}+\left(\frac{4}{5}\right)^5\times\frac{1}{5}\right)\) | M1 | From final answer. Condone omission of \(\left(\frac{4}{5}\right)^5\times\frac{1}{5}\) |
| \(\frac{4096}{15625}\), \(0.262\) | A1 | \(0.262144\) rounded to more than 3SF |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{10}{15}\times\frac{9}{14}\times\frac{8}{13}\) | M1 | \(\frac{a}{15}\times\frac{a-1}{14}\times\frac{a-2}{13}\), no additional terms |
| \(\frac{24}{91}\), \(0.264\) | A1 | \(0.263736\) rounded to more than 3SF |
| Alternative method 1: | ||
| \(\frac{3}{15}\times\frac{2}{14}\times\frac{1}{13}+3\times\frac{3}{15}\times\frac{2}{14}\times\frac{7}{13}+3\times\frac{3}{15}\times\frac{7}{14}\times\frac{6}{13}+\frac{7}{15}\times\frac{6}{14}\times\frac{5}{13}\) | M1 | \([3\text{Ls} + 2\text{Ls1S} + 1\text{L2Ss} + 3\text{Ss}]\). Condone one numerator error. Condone no multiplications seen if tree diagram complete with probabilities on each branch, scenarios listed and attempt at evaluation |
| \(\frac{24}{91}\), \(0.264\) | A1 | \(0.263736\) rounded to more than 3SF |
| Alternative method 2: | ||
| \(1-\left(\frac{5}{15}\times\frac{4}{14}\times\frac{3}{13}+3\times\frac{5}{15}\times\frac{4}{14}\times\frac{10}{13}+3\times\frac{5}{15}\times\frac{10}{14}\times\frac{9}{13}\right)\) | M1 | \(1 - P(3,2,1 \text{ oranges})\). Condone one numerator error |
| \(\frac{24}{91}\), \(0.264\) | A1 | \(0.263736\) rounded to more than 3SF |
| Alternative method 3: | ||
| \(\dfrac{^{10}C_3}{^{15}C_3}\) | M1 | |
| \(\frac{24}{91}\), \(0.264\) | A1 | \(0.263736\) rounded to more than 3SF |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{7}{15}\times\frac{5}{14}\times\frac{3}{13}\times 3!\) | M1 | All probabilities of the form: \(\frac{7}{a}\times\frac{5}{b}\times\frac{3}{c}\), \(13 \leqslant a,b,c \leqslant 15\) |
| M1 | \(\frac{e}{f}\times\frac{g}{h}\times\frac{i}{j}\times 3!\) where \(e,f,g,h,i,j\) positive integers forming probabilities, or 6 identical probability calculations or values added, no additional terms | |
| \(\frac{3}{13}\), \(0.231\) | A1 | \(0.230769\) rounded (not truncated) to more than 3SF |
| Alternative method: | ||
| \(\dfrac{^3C_1 \times\, ^5C_1 \times\, ^7C_1}{^{15}C_3}\) | M1 | \(\dfrac{^3C_1 \times\, ^5C_1 \times\, ^7C_1}{k}\), \(k\) integer \(> 1\). Condone use of permutations |
| M1 | \(\dfrac{^3C_a \times\, ^5C_b \times\, ^7C_c}{^{15}C_3}\), \(0{<}a{<}3\), \(0{<}b{<}5\), \(0{<}c{<}7\). Condone use of permutations | |
| \(\frac{3}{13}\), \(0.231\) | A1 | \(0.230769\) rounded (not truncated) to more than 3SF |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{7}{15}\times\frac{6}{14}\times\frac{5}{13}+\frac{3}{15}\times\frac{7}{14}\times\frac{6}{13}\times 3\) [= \(\frac{14}{65}+\frac{24}{91}\)] divided by *their*(c) | B1 | \(\frac{3}{15}\times\frac{7}{14}\times\frac{6}{13}\times 3\) seen (SSL, SLS, LSS). SC B1 \(\frac{3}{65}\times 3\), \(\frac{126}{2730}\times 3\) seen |
| B1 | \(\frac{7}{15}\times\frac{6}{14}\times\frac{5}{13}\) seen in numerator (SSS). SCB1 \(\frac{210}{2730}\), \(\frac{1}{13}\) seen in numerator | |
| M1 | Fraction with *their* (c) or correct in denominator \(\left(\frac{720}{2730}, \frac{24}{91}, 0.263736\right)\) | |
| \(= \frac{49}{60}\), \(0.817\) | A1 | Accept \(0.81\dot{6}\) |
| Alternative method: | ||
| \(\dfrac{^7C_2 \times\, ^3C_1 +\, ^7C_3}{^{10}C_3}\) | B1 | \(^7C_2 \times\, ^3C_1\) seen (SSL, SLS, LSS). SCB1 \(21\times 3\) seen or use of permutations |
| B1 | \(^7C_3\) seen in numerator (SSS). SCB1 \(35\) seen in numerator or use of permutations | |
| M1 | Fraction with \(^{10}C_3\) or consistent with *their* numerator of 6(c) in denominator | |
| \(= \frac{49}{60}\), \(0.817\) | A1 | Accept \(0.81\dot{6}\) |
| 4 |
## Question 6:
### Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| [Probability of lemon $= \frac{3}{15} = \frac{1}{5}$] | | |
| $\left(\frac{4}{5}\right)^6 \times \frac{1}{5} = \frac{4096}{78125}$, $0.0524$ | **B1** | $0.0524288$ rounded to more than 3SF if final answer |
| | **1** | |
---
### Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\left(1-\frac{1}{5}\right)^6$ | **M1** | or $\left(\frac{4}{5}\right)^6$. FT *their* $\frac{1}{5}$ or correct. From final answer. Condone $\left(\frac{4}{5}\right)^5$ or $\left(\frac{1}{5}\right)\times\left(\frac{4}{5}\right)^5 + \left(\frac{4}{5}\right)^6$ |
| $\frac{4096}{15625}$, $0.262$ | **A1** | $0.262144$ rounded to more than 3SF |
| **Alternative method:** | | |
| $[1 - P(1,2,3,4,5,[6]) =]$ $1-\left(\frac{1}{5}+\frac{4}{5}\times\frac{1}{5}+\left(\frac{4}{5}\right)^2\times\frac{1}{5}+\left(\frac{4}{5}\right)^3\times\frac{1}{5}+\left(\frac{4}{5}\right)^4\times\frac{1}{5}+\left(\frac{4}{5}\right)^5\times\frac{1}{5}\right)$ | **M1** | From final answer. Condone omission of $\left(\frac{4}{5}\right)^5\times\frac{1}{5}$ |
| $\frac{4096}{15625}$, $0.262$ | **A1** | $0.262144$ rounded to more than 3SF |
| | **2** | |
---
### Part (c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{10}{15}\times\frac{9}{14}\times\frac{8}{13}$ | **M1** | $\frac{a}{15}\times\frac{a-1}{14}\times\frac{a-2}{13}$, no additional terms |
| $\frac{24}{91}$, $0.264$ | **A1** | $0.263736$ rounded to more than 3SF |
| **Alternative method 1:** | | |
| $\frac{3}{15}\times\frac{2}{14}\times\frac{1}{13}+3\times\frac{3}{15}\times\frac{2}{14}\times\frac{7}{13}+3\times\frac{3}{15}\times\frac{7}{14}\times\frac{6}{13}+\frac{7}{15}\times\frac{6}{14}\times\frac{5}{13}$ | **M1** | $[3\text{Ls} + 2\text{Ls1S} + 1\text{L2Ss} + 3\text{Ss}]$. Condone one numerator error. Condone no multiplications seen if tree diagram complete with probabilities on each branch, scenarios listed and attempt at evaluation |
| $\frac{24}{91}$, $0.264$ | **A1** | $0.263736$ rounded to more than 3SF |
| **Alternative method 2:** | | |
| $1-\left(\frac{5}{15}\times\frac{4}{14}\times\frac{3}{13}+3\times\frac{5}{15}\times\frac{4}{14}\times\frac{10}{13}+3\times\frac{5}{15}\times\frac{10}{14}\times\frac{9}{13}\right)$ | **M1** | $1 - P(3,2,1 \text{ oranges})$. Condone one numerator error |
| $\frac{24}{91}$, $0.264$ | **A1** | $0.263736$ rounded to more than 3SF |
| **Alternative method 3:** | | |
| $\dfrac{^{10}C_3}{^{15}C_3}$ | **M1** | |
| $\frac{24}{91}$, $0.264$ | **A1** | $0.263736$ rounded to more than 3SF |
| | **2** | |
---
### Part (d):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{7}{15}\times\frac{5}{14}\times\frac{3}{13}\times 3!$ | **M1** | All probabilities of the form: $\frac{7}{a}\times\frac{5}{b}\times\frac{3}{c}$, $13 \leqslant a,b,c \leqslant 15$ |
| | **M1** | $\frac{e}{f}\times\frac{g}{h}\times\frac{i}{j}\times 3!$ where $e,f,g,h,i,j$ positive integers forming probabilities, or 6 identical probability calculations or values added, no additional terms |
| $\frac{3}{13}$, $0.231$ | **A1** | $0.230769$ rounded (not truncated) to more than 3SF |
| **Alternative method:** | | |
| $\dfrac{^3C_1 \times\, ^5C_1 \times\, ^7C_1}{^{15}C_3}$ | **M1** | $\dfrac{^3C_1 \times\, ^5C_1 \times\, ^7C_1}{k}$, $k$ integer $> 1$. Condone use of permutations |
| | **M1** | $\dfrac{^3C_a \times\, ^5C_b \times\, ^7C_c}{^{15}C_3}$, $0{<}a{<}3$, $0{<}b{<}5$, $0{<}c{<}7$. Condone use of permutations |
| $\frac{3}{13}$, $0.231$ | **A1** | $0.230769$ rounded (not truncated) to more than 3SF |
| | **3** | |
---
### Part (e):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{7}{15}\times\frac{6}{14}\times\frac{5}{13}+\frac{3}{15}\times\frac{7}{14}\times\frac{6}{13}\times 3$ [= $\frac{14}{65}+\frac{24}{91}$] divided by *their*(c) | **B1** | $\frac{3}{15}\times\frac{7}{14}\times\frac{6}{13}\times 3$ seen (SSL, SLS, LSS). **SC B1** $\frac{3}{65}\times 3$, $\frac{126}{2730}\times 3$ seen |
| | **B1** | $\frac{7}{15}\times\frac{6}{14}\times\frac{5}{13}$ seen in numerator (SSS). **SCB1** $\frac{210}{2730}$, $\frac{1}{13}$ seen in numerator |
| | **M1** | Fraction with *their* **(c)** or correct in denominator $\left(\frac{720}{2730}, \frac{24}{91}, 0.263736\right)$ |
| $= \frac{49}{60}$, $0.817$ | **A1** | Accept $0.81\dot{6}$ |
| **Alternative method:** | | |
| $\dfrac{^7C_2 \times\, ^3C_1 +\, ^7C_3}{^{10}C_3}$ | **B1** | $^7C_2 \times\, ^3C_1$ seen (SSL, SLS, LSS). **SCB1** $21\times 3$ seen or use of permutations |
| | **B1** | $^7C_3$ seen in numerator (SSS). **SCB1** $35$ seen in numerator or use of permutations |
| | **M1** | Fraction with $^{10}C_3$ or consistent with *their* numerator of **6(c)** in denominator |
| $= \frac{49}{60}$, $0.817$ | **A1** | Accept $0.81\dot{6}$ |
| | **4** | |6 A factory produces chocolates in three flavours: lemon, orange and strawberry in the ratio $3 : 5 : 7$ respectively. Nell checks the chocolates on the production line by choosing chocolates randomly one at a time.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that the first chocolate with lemon flavour that Nell chooses is the 7th chocolate that she checks.
\item Find the probability that the first chocolate with lemon flavour that Nell chooses is after she has checked at least 6 chocolates.\\
'Surprise' boxes of chocolates each contain 15 chocolates: 3 are lemon, 5 are orange and 7 are strawberry.
Petra has a box of Surprise chocolates. She chooses 3 chocolates at random from the box. She eats each chocolate before choosing the next one.
\item Find the probability that none of Petra's 3 chocolates has orange flavour.
\item Find the probability that each of Petra's 3 chocolates has a different flavour.
\item Find the probability that at least 2 of Petra's 3 chocolates have strawberry flavour given that none of them has orange flavour.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2022 Q6 [12]}}