| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2010 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Definitions |
| Type | Sequential events and tree diagrams |
| Difficulty | Standard +0.3 This is a straightforward probability question requiring calculation of individual probabilities from given information, then applying basic probability rules (multiplication for independent events, addition for mutually exclusive outcomes). The arithmetic involves simple fractions and the logic is standard for S1 level, making it slightly easier than average but requiring careful organization across multiple cases. |
| Spec | 2.03a Mutually exclusive and independent events2.03b Probability diagrams: tree, Venn, sample space |
| Answer | Marks | Guidance |
|---|---|---|
| (i) | M1 | Obtaining probs of each person for each entrance (can be implied or awarded in part (i) or part (ii)) |
| \(A\) | \(B\) | \(C\) |
| Rick | 1/3 | 2/9 |
| Brenda | 1/4 | 1/4 |
| Ali | 2/35 | 2/35 |
| \(P(\text{Rick } B, \text{ Brenda } B, \text{ Ali not } B) = P(\text{Rick } B, \text{ Brenda not } B, \text{ Ali } B)\) | M1 | Considering options 2 meet 1 doesn't, must have at least two 3-factor terms |
| \(+ P(\text{Rick not } B, \text{ Brenda } B, \text{ Ali } B) = 11/210 + 2/210 + 1/90 = 23/315\) | ||
| \(P(\text{Rick } B, \text{ Brenda } B, \text{ Ali } B) = 1/315\) | M1 | Adding option all three meet, must be added to a prob |
| \(\text{Prob(at least 2 at entrance } B) = 24/315 \text{ (8/105) (0.0762)}\) | A1 | Correct answer |
| [4] | ||
| (ii) \(P(\text{entrance } A) = 1/210 \text{ (0.00476)}\) | M1 | Obtaining a three-factor prob for any entrance |
| \(P(\text{entrance } B) = 1/315 \text{ (0.00317)}\) | M1 | Adding four three-factor probabilities for the 4 entrances |
| \(P(\text{entrance } C) = 1/63 \text{ (0.0159)}\) | ||
| \(P(\text{entrance } D) = 1/30 \text{ (0.0333)}\) | ||
| \(P(\text{same entrance}) = 2/35 \text{ (0.0571)}\) | A1 | Two or more correct entrance probabilities |
| A1 | Correct answer | |
| [4] |
**(i)** | M1 | Obtaining probs of each person for each entrance (can be implied or awarded in part (i) or part (ii))
| | $A$ | $B$ | $C$ | $D$ |
|---|---|---|---|---|
| Rick | 1/3 | 2/9 | 2/9 | 2/9 |
| Brenda | 1/4 | 1/4 | 1/4 | 1/4 |
| Ali | 2/35 | 2/35 | 2/7 | 3/5 |
$P(\text{Rick } B, \text{ Brenda } B, \text{ Ali not } B) = P(\text{Rick } B, \text{ Brenda not } B, \text{ Ali } B)$ | M1 | Considering options 2 meet 1 doesn't, must have at least two 3-factor terms
$+ P(\text{Rick not } B, \text{ Brenda } B, \text{ Ali } B) = 11/210 + 2/210 + 1/90 = 23/315$ |
$P(\text{Rick } B, \text{ Brenda } B, \text{ Ali } B) = 1/315$ | M1 | Adding option all three meet, must be added to a prob
$\text{Prob(at least 2 at entrance } B) = 24/315 \text{ (8/105) (0.0762)}$ | A1 | Correct answer
| [4] |
**(ii)** $P(\text{entrance } A) = 1/210 \text{ (0.00476)}$ | M1 | Obtaining a three-factor prob for any entrance
$P(\text{entrance } B) = 1/315 \text{ (0.00317)}$ | M1 | Adding four three-factor probabilities for the 4 entrances
$P(\text{entrance } C) = 1/63 \text{ (0.0159)}$ |
$P(\text{entrance } D) = 1/30 \text{ (0.0333)}$ |
$P(\text{same entrance}) = 2/35 \text{ (0.0571)}$ | A1 | Two or more correct entrance probabilities
| A1 | Correct answer
| [4] |5 Three friends, Rick, Brenda and Ali, go to a football match but forget to say which entrance to the ground they will meet at. There are four entrances, $A , B , C$ and $D$. Each friend chooses an entrance independently.
\begin{itemize}
\item The probability that Rick chooses entrance $A$ is $\frac { 1 } { 3 }$. The probabilities that he chooses entrances $B , C$ or $D$ are all equal.
\item Brenda is equally likely to choose any of the four entrances.
\item The probability that Ali chooses entrance $C$ is $\frac { 2 } { 7 }$ and the probability that he chooses entrance $D$ is $\frac { 3 } { 5 }$. The probabilities that he chooses the other two entrances are equal.\\
(i) Find the probability that at least 2 friends will choose entrance $B$.\\
(ii) Find the probability that the three friends will all choose the same entrance.
\end{itemize}
\hfill \mbox{\textit{CAIE S1 2010 Q5 [8]}}