| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2024 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Conditional probability with normal |
| Difficulty | Standard +0.8 This question combines normal distribution standardisation (routine) with conditional probability and independent events. Part (a) is straightforward z-score calculation. Part (b) requires careful interpretation: the athlete must fail first throw, then succeed on second throw with distance >44m (requiring a second z-score calculation), all conditioned on eventually qualifying. The multi-step conditional probability reasoning and need to identify the correct event structure elevates this above standard S1 questions. |
| Spec | 2.03a Mutually exclusive and independent events2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X < 38.8) = P\!\left(Z < \frac{38.8-40}{4}\right)[=P(Z<-0.3)]\) | M1 | For standardising using 38.8, 40 and 4 (allow \(\pm\)) |
| \(= 1 - 0.6179 = 0.3821\) | A1* | Must see \(1-0.6179\) or \(0.38209\) or \(0.38208\ldots\) or better; answer is given so no incorrect working can be seen |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(\text{Qualify}) = 1-(0.3821)^3\) or \(1-0.3821+0.3821\times(1-0.3821)+0.3821^2\times(1-0.3821)\) \([=0.9442]\) | M1 | For correct method to find probability of qualifying |
| \(P(X>44) = P\!\left(Z > \frac{44-40}{4}\right)[=P(Z>1)]\) | M1 | For standardising using 44, 40 and 4 (implied by \(1-0.8413\) or awrt 0.1587) |
| \([=1-0.8413]=0.1587\) | A1 | awrt 0.16 |
| \(P(X>44 \text{ on 2nd attempt} | \text{Qualify}) = \dfrac{0.3821\times 0.1587}{0.9442}\) | M1 |
| \(0.06422\ldots\) awrt 0.0642 | A1 |
## Question 5:
### Part (a):
$P(X < 38.8) = P\!\left(Z < \frac{38.8-40}{4}\right)[=P(Z<-0.3)]$ | M1 | For standardising using 38.8, 40 and 4 (allow $\pm$)
$= 1 - 0.6179 = 0.3821$ | A1* | Must see $1-0.6179$ or $0.38209$ or $0.38208\ldots$ or better; answer is given so no incorrect working can be seen
### Part (b):
$P(\text{Qualify}) = 1-(0.3821)^3$ or $1-0.3821+0.3821\times(1-0.3821)+0.3821^2\times(1-0.3821)$ $[=0.9442]$ | M1 | For correct method to find probability of qualifying
$P(X>44) = P\!\left(Z > \frac{44-40}{4}\right)[=P(Z>1)]$ | M1 | For standardising using 44, 40 and 4 (implied by $1-0.8413$ or awrt 0.1587)
$[=1-0.8413]=0.1587$ | A1 | awrt 0.16
$P(X>44 \text{ on 2nd attempt}|\text{Qualify}) = \dfrac{0.3821\times 0.1587}{0.9442}$ | M1 | For correct ratio of probabilities ft their 0.1587 and their 0.9442; use of 0.6179 in denominator is M0
$0.06422\ldots$ awrt 0.0642 | A1 |
---\begin{enumerate}
\item The distance an athlete can throw a discus is normally distributed with mean 40 m and standard deviation 4 m\\
(a) Using standardisation, show that the probability that this athlete throws the discus less than 38.8 m is 0.3821
\end{enumerate}
This athlete enters a discus competition.\\
To qualify for the final, they have 3 attempts to throw the discus a distance of more than 38.8 m\\
Once they qualify, they do not use any of their remaining attempts.\\
Given that they qualified for the final and that throws are independent,\\
(b) find the probability that this athlete qualified for the final on their second throw with a distance of more than 44 m
\hfill \mbox{\textit{Edexcel S1 2024 Q5 [7]}}