Moderate -0.5 This is a straightforward application of the independence definition P(R∩T) = P(R)×P(T) with simple counting on a 6×6 sample space. The main work is systematically listing outcomes for each event, which requires care but no sophisticated reasoning or novel insight—slightly easier than average.
2 Ashfaq throws two fair dice and notes the numbers obtained. \(R\) is the event 'The product of the two numbers is 12 '. \(T\) is the event 'One of the numbers is odd and one of the numbers is even'. By finding appropriate probabilities, determine whether events \(R\) and \(T\) are independent.
As \(P(R) \times P(T) = P(R \cap T)\) OR as \(P(R\
T) = P(R)\)
M1
The events are independent.
A1
Correct conclusion must have all probs correct
## Question 2:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(R) = 4/36 = 1/9$ | M1 | Attempt at $P(R)$ by probability space diagram or listing more than half the options; must see a prob, just a list is not enough |
| $P(T) = P(O,E) + P(E,O) = 1/4 + 1/4 = 1/2$ OR $P(R\|T) = 1/9$ | M1 | Attempt at $P(T)$ or $P(R\|T)$ involving more than half the options |
| $P(R \cap T) = P(3,4) + P(4,3) = 2/36 = 1/18$ OR $P(R\|T) = 1/9$ | B1 | Value stated, not from $P(R) \times P(T)$ e.g. from probability space diagram |
| As $P(R) \times P(T) = P(R \cap T)$ OR as $P(R\|T) = P(R)$ | M1 | Comparing product values with $P(R \cap T)$, or comparing $P(R\|T)$ with $P(R)$ |
| The events are independent. | A1 | Correct conclusion must have all probs correct |
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2 Ashfaq throws two fair dice and notes the numbers obtained. $R$ is the event 'The product of the two numbers is 12 '. $T$ is the event 'One of the numbers is odd and one of the numbers is even'. By finding appropriate probabilities, determine whether events $R$ and $T$ are independent.\\
\hfill \mbox{\textit{CAIE S1 2017 Q2 [5]}}