| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/3 (Pre-U Mathematics Paper 3) |
| Year | 2012 |
| Session | June |
| Marks | 10 |
| Topic | Conditional Probability |
| Type | Conditional with three or more stages |
| Difficulty | Moderate -0.8 This is a straightforward conditional probability question using a tree diagram with clearly stated probabilities at each stage. The calculations involve basic multiplication and addition of probabilities along branches, with part (iv) requiring simple application of conditional probability formula P(A|B) = P(A∩B)/P(B). All steps are routine and mechanical with no conceptual challenges beyond standard A-level probability techniques. |
| Spec | 2.03b Probability diagrams: tree, Venn, sample space2.03c Conditional probability: using diagrams/tables |
**(i)** Their product of three fails $(0.55 \times 0.4 \times 0.85)$ — M1
Obtain 0.187 — A1 [2]
**(ii)** Attempt $P(F)P(S)$ $(0.55 \times 0.6 = 0.33)$ — M1
Attempt $P(F)P(F)P(S)$ $(0.55 \times 0.4 \times 0.15 = 0.033)$ — M1
Or $1 - (0.45 + \textbf{(i)})$
Obtain 0.363 — A1 [3]
**(iii)** Use $P(S)$ + answer to **(ii)** — M1
Or $1 -$ **(i)**
Obtain 0.813 — A1 [2]
**(iv)** Attempt to divide two probabilities — M1
Divide their $P(F)P(S)$ by their **(iii)** — M1
Obtain 0.406 (or 110/271) — A1 [3]
**Total: [10]**
*ft **(i)** if appropriate; ft **(i)** or **(ii)** as appropriate; ft **(iii)**.*5 In an archery competition, competitors are allowed up to three attempts to hit the bulls-eye. No one who succeeds may try again.\\
$45 \%$ of those entering the competition hit the bulls-eye first time. For those who fail to hit it the first time, $60 \%$ of those attempting it for the second time succeed in hitting it. For those who fail twice, only $15 \%$ of those attempting it for the third time succeed in hitting it. By drawing a tree diagram, or otherwise,\\
(i) find the probability that a randomly chosen competitor fails at all three attempts,\\
(ii) find the probability that a randomly chosen competitor fails at the first attempt but succeeds at either the second or third attempt,\\
(iii) find the probability that a randomly chosen competitor succeeds in hitting the bulls-eye,\\
(iv) find the probability that a randomly chosen competitor requires exactly two attempts given that the competitor is successful.
\hfill \mbox{\textit{Pre-U Pre-U 9794/3 2012 Q5 [10]}}