| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2021 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Two unknowns from sum and expectation |
| Difficulty | Moderate -0.5 This is a standard two-equation, two-unknown problem in discrete probability. Students must use the sum of probabilities equals 1 and the given condition P(X≥0)=3P(X<0) to form simultaneous equations. The algebra is straightforward with no conceptual challenges beyond basic S1 probability distribution properties, making it slightly easier than average. |
| Spec | 2.04a Discrete probability distributions |
| \(x\) | - 2 | - 1 | 0 | 1 | 2 |
| \(\mathrm { P } ( X = x )\) | \(p\) | \(p\) | 0.1 | \(q\) | \(q\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(p + p + 0.1 + q + q = 1\) | B1 | Sum of probabilities = 1 |
| \(0.1 + 2q = 3(2p)\) | B1 | Use given information |
| Attempt to solve two correct equations in \(p\) and \(q\) | M1 | Either substitution to form single equation in \(p\) or \(q\); Or elimination method writing both equations in form \(ap + bq = c\) and +/− to find equation in one unknown |
| \(p = \frac{1}{8}\) or 0.125 and \(q = \frac{13}{40}\) or 0.325 | A1 | CAO, both WWW |
## Question 2:
| Answer | Mark | Guidance |
|--------|------|----------|
| $p + p + 0.1 + q + q = 1$ | B1 | Sum of probabilities = 1 |
| $0.1 + 2q = 3(2p)$ | B1 | Use given information |
| Attempt to solve two correct equations in $p$ and $q$ | M1 | Either substitution to form single equation in $p$ or $q$; Or elimination method writing both equations in form $ap + bq = c$ and +/− to find equation in one unknown |
| $p = \frac{1}{8}$ or 0.125 and $q = \frac{13}{40}$ or 0.325 | A1 | CAO, both WWW |
---2 The random variable $X$ can take only the values $- 2 , - 1,0,1,2$. The probability distribution of $X$ is given in the following table.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & - 2 & - 1 & 0 & 1 & 2 \\
\hline
$\mathrm { P } ( X = x )$ & $p$ & $p$ & 0.1 & $q$ & $q$ \\
\hline
\end{tabular}
\end{center}
Given that $\mathrm { P } ( X \geqslant 0 ) = 3 \mathrm { P } ( X < 0 )$, find the values of $p$ and $q$.\\
\hfill \mbox{\textit{CAIE S1 2021 Q2 [4]}}