| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2018 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Verify conditions in context |
| Difficulty | Moderate -0.8 This is a straightforward S2 binomial distribution question requiring only standard recall and basic calculations. Part (a) tests knowledge of binomial conditions (textbook recall), parts (b)-(d) involve routine probability calculations using the binomial formula with small n=6, requiring no problem-solving insight or complex manipulation. |
| Spec | 2.04b Binomial distribution: as model B(n,p)5.02d Binomial: mean np and variance np(1-p) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Only 2 outcomes Heads and Tails oe | B1 | A correct statement – does not need to be in context |
| Constant probability of spinning a Head/Tail oe | B1 | A second correct statement in context; include coin or heads or tails (do not allow H and T) or spins/flip oe |
| Coin is spun a fixed number of times oe | ||
| Each spin of the coin is independent oe |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(T \sim B(6, 0.5)\) | ||
| \(P(T \leq 5) - P(T \leq 4) = 0.9844 - 0.8906\) or \(6\left(\frac{1}{2}\right)^5\left(\frac{1}{2}\right)\) oe | M1 | Writing or using \(B(6, 0.5)\) and writing or using \(P(T \leq 5) - P(T \leq 4)\) or \(\left[6\left(\frac{1}{2}\right)^6\right]\) oe |
| \(= 0.09375\) or \(\frac{3}{32}\) oe | A1 | awrt 0.0938 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P(T = 4,5,6) = 1 - P(T \leq 3)\) | M1 | For realising they need find \(P(T=4, 5\) or \(6)\) eg \(1-P(T \leq 3)\) or \(P(T \geq 4)\) |
| \(= 1 - 0.6563\) | ||
| \(= 0.3437\) or \(\frac{11}{32}\) | A1 | awrt 0.344 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P(H = 3,4,5,6) = 1 - P(H \leq 2)\) | B1, M1d | B1: writing/using \(B(6, 0.25)\) and \(P(H \geq 3)\) oe or writing/using \(B(6, 0.75)\) and \(P(T \leq 3)\); M1d dep on B1 for \(1 - P(H \leq 2)\) |
| \(= 1 - 0.8306\) | dep on B1: \((0.25)^6 + 6(0.75)(0.25)^5 + 15(0.75)^2(0.25)^4 + 20(0.75)^3(0.25)^3\) | |
| \(= 0.1694\) or \(\frac{347}{2048}\) | A1 | awrt 0.169; awrt 0.169 with no incorrect working gains B1M1A1; Only accept correct use of H and T in probability statement unless variable correctly defined |
# Question 2:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Only 2 outcomes **Heads** and **Tails** oe | B1 | A correct statement – does not need to be in context |
| Constant probability of **spinning** a **Head/Tail** oe | B1 | A second correct statement in context; include coin or heads or tails (do not allow H and T) or spins/flip oe |
| Coin is **spun** a fixed number of times oe | | |
| Each **spin** of the **coin** is independent oe | | |
## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $T \sim B(6, 0.5)$ | | |
| $P(T \leq 5) - P(T \leq 4) = 0.9844 - 0.8906$ or $6\left(\frac{1}{2}\right)^5\left(\frac{1}{2}\right)$ oe | M1 | Writing or using $B(6, 0.5)$ **and** writing or using $P(T \leq 5) - P(T \leq 4)$ or $\left[6\left(\frac{1}{2}\right)^6\right]$ oe |
| $= 0.09375$ or $\frac{3}{32}$ oe | A1 | awrt 0.0938 |
## Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P(T = 4,5,6) = 1 - P(T \leq 3)$ | M1 | For realising they need find $P(T=4, 5$ or $6)$ eg $1-P(T \leq 3)$ or $P(T \geq 4)$ |
| $= 1 - 0.6563$ | | |
| $= 0.3437$ or $\frac{11}{32}$ | A1 | awrt 0.344 |
## Part (d)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P(H = 3,4,5,6) = 1 - P(H \leq 2)$ | B1, M1d | B1: writing/using $B(6, 0.25)$ **and** $P(H \geq 3)$ oe **or** writing/using $B(6, 0.75)$ **and** $P(T \leq 3)$; M1d dep on B1 for $1 - P(H \leq 2)$ |
| $= 1 - 0.8306$ | | dep on B1: $(0.25)^6 + 6(0.75)(0.25)^5 + 15(0.75)^2(0.25)^4 + 20(0.75)^3(0.25)^3$ |
| $= 0.1694$ or $\frac{347}{2048}$ | A1 | awrt 0.169; **awrt 0.169 with no incorrect working gains B1M1A1**; Only accept correct use of H and T in probability statement unless variable correctly defined |2. A fair coin is spun 6 times and the random variable $T$ represents the number of tails obtained.
\begin{enumerate}[label=(\alph*)]
\item Give two reasons why a binomial model would be a suitable distribution for modelling $T$.
\item Find $\mathrm { P } ( T = 5 )$
\item Find the probability of obtaining more tails than heads.
A second coin is biased such that the probability of obtaining a head is $\frac { 1 } { 4 }$ This second coin is spun 6 times.
\item Find the probability that, for the second coin, the number of heads obtained is greater than or equal to the number of tails obtained.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2018 Q2 [9]}}