Moderate -0.3 This is a straightforward application of the independence definition P(S∩T) = P(S)P(T) using systematic enumeration of outcomes from two dice. While it requires careful counting and organization, it's a standard textbook exercise with no conceptual subtlety—slightly easier than average due to the mechanical nature of the task.
1 Two ordinary fair dice are thrown and the numbers obtained are noted. Event \(S\) is 'The sum of the numbers is even'. Event \(T\) is 'The sum of the numbers is either less than 6 or a multiple of 4 or both'. Showing your working, determine whether the events \(S\) and \(T\) are independent.
\(P(S \cap T)\) found by multiplication scores M0; M1 awarded if *their* value is identifiable in their sample space diagram or Venn diagram or list of terms or probability distribution table (oe)
\(P(S) \cdot P(T) \neq P(S \cap T)\) so not independent
A1
\(\frac{8}{36}\), \(\frac{10}{36}\): \(P(S) \times P(T)\) and \(P(S \cap T)\) seen in workings and correct conclusion stated, www
\(P(S \cap T)\) found by multiplication scores M0; M1 awarded if *their* value is identifiable in their sample space diagram or Venn diagram or list of terms or probability distribution table (oe)
\(P(S\
T) = \frac{10}{16}\) or \(P(T\
S) = \frac{10}{18}\); \(P(S\
Total
4
## Question 1:
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(S) = \frac{1}{2}$ | B1 | |
| $P(T) = \frac{16}{36} \left(\frac{4}{9}\right)$ | B1 | |
| $P(S \cap T) = \frac{10}{36} \left(\frac{5}{18}\right)$ | M1 | $P(S \cap T)$ found by multiplication scores M0; M1 awarded if *their* value is identifiable in their sample space diagram **or** Venn diagram **or** list of terms **or** probability distribution table (oe) |
| $P(S) \cdot P(T) \neq P(S \cap T)$ so not independent | A1 | $\frac{8}{36}$, $\frac{10}{36}$: $P(S) \times P(T)$ and $P(S \cap T)$ seen in workings and correct conclusion stated, www |
**Alternative method for Question 1:**
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(S) = \frac{1}{2}$ | B1 | |
| $P(T) = \frac{16}{36} \left(\frac{4}{9}\right)$ | B1 | |
| $P(S \cap T) = \frac{10}{36} \left(\frac{5}{18}\right)$ | M1 | $P(S \cap T)$ found by multiplication scores M0; M1 awarded if *their* value is identifiable in their sample space diagram **or** Venn diagram **or** list of terms **or** probability distribution table (oe) |
| $P(S\|T) = \frac{10}{16}$ or $P(T\|S) = \frac{10}{18}$; $P(S\|T) \neq P(S)$ or $P(T\|S) \neq P(T)$ so not independent | A1 | **Either** $\frac{18}{36}$, $\frac{10}{16}$, $P(S)$ and $P(S\|T)$ seen in workings and correct conclusion stated, www **Or** $\frac{16}{36}$, $\frac{10}{18}$, $P(T)$ and $P(T\|S)$ seen in workings and correct conclusion stated, www |
| **Total** | **4** | |
1 Two ordinary fair dice are thrown and the numbers obtained are noted. Event $S$ is 'The sum of the numbers is even'. Event $T$ is 'The sum of the numbers is either less than 6 or a multiple of 4 or both'. Showing your working, determine whether the events $S$ and $T$ are independent.\\
\hfill \mbox{\textit{CAIE S1 2019 Q1 [4]}}