| Exam Board | Edexcel |
|---|---|
| Module | Paper 3 (Paper 3) |
| Year | 2018 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Verify conditions in context |
| Difficulty | Moderate -0.3 This is a straightforward binomial distribution question testing standard concepts: stating assumptions (routine recall), calculating P(H≥4) using tables/calculator, finding geometric distribution probability P(F=5), and finding a constant α from a simple linear probability model. Part (f) requires basic comparison of models. All parts are textbook-standard with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04a Discrete probability distributions2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| The probability of a dart hitting the target is constant (from child to child and for each throw by each child) | B1 | For stating probability (or possibility or chance) is constant (or fixed or same) |
| The throws of each of the darts are independent | B1 | For stating throws are independent ["trials" are independent is B0] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(H \geq 4) = 1 - P(H \leq 3) = 1 - 0.9872 = 0.012795\ldots\) awrt \(\mathbf{0.0128}\) | B1 | For awrt 0.0128 (found on calculator) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(F=5) = 0.9^4 \times 0.1 = 0.06561\) = awrt \(\mathbf{0.0656}\) | M1, A1 | M1 for probability expression of form \((1-p)^4 \times p\) where \(0 < p < 1\); A1 for awrt 0.0656. Allow M1A0 for answer only of 0.066 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Table with \(n = 1, 2, \ldots, 10\) and \(P(F=n) = 0.01,\ 0.01+\alpha, \ldots,\ 0.01+9\alpha\) | M1 | 1st M1 for setting up distribution of \(F\) with at least 3 correct values of \(n\) and \(P(F=n)\) in terms of \(\alpha\) |
| Sum of probs \(= 1 \Rightarrow \frac{10}{2}[2\times0.01 + 9\alpha] = 1\) | M1A1 | 2nd M1 for use of sum of probs \(= 1\) and clear summation or arithmetic series formula (allow 1 error or missing term) |
| \(5(0.02 + 9\alpha) = 1\) or \(0.1 + 45\alpha = 1\), so \(\alpha = \mathbf{0.02}\) | A1 | \(\alpha = 0.02\) must be exact and come from correct working |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(F=5 \mid \text{Thomas' model}) = \mathbf{0.09}\) | B1ft | For value resulting from \(0.01 + 4\times\text{"their }\alpha\text{"}\). If answer same as (c) or rounded version of (c), score B0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Peta's model assumes the probability of hitting target is constant and Thomas' model assumes this probability increases with each attempt | B1 | Suitable comment about the probability of hitting the target. Allow idea that Peta's model suggests the dart may never hit the target but Thomas' says it will hit at least once (in first 10 throws) |
# Question 3:
## Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| The probability of a dart hitting the target is constant (from child to child and for each throw by each child) | B1 | For stating probability (or possibility or chance) is constant (or fixed or same) |
| The throws of each of the darts are independent | B1 | For stating throws are independent ["trials" are independent is B0] |
## Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(H \geq 4) = 1 - P(H \leq 3) = 1 - 0.9872 = 0.012795\ldots$ awrt $\mathbf{0.0128}$ | B1 | For awrt 0.0128 (found on calculator) |
## Part (c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(F=5) = 0.9^4 \times 0.1 = 0.06561$ = awrt $\mathbf{0.0656}$ | M1, A1 | M1 for probability expression of form $(1-p)^4 \times p$ where $0 < p < 1$; A1 for awrt 0.0656. Allow M1A0 for answer only of 0.066 |
## Part (d):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Table with $n = 1, 2, \ldots, 10$ and $P(F=n) = 0.01,\ 0.01+\alpha, \ldots,\ 0.01+9\alpha$ | M1 | 1st M1 for setting up distribution of $F$ with at least 3 correct values of $n$ and $P(F=n)$ in terms of $\alpha$ |
| Sum of probs $= 1 \Rightarrow \frac{10}{2}[2\times0.01 + 9\alpha] = 1$ | M1A1 | 2nd M1 for use of sum of probs $= 1$ and clear summation or arithmetic series formula (allow 1 error or missing term) |
| $5(0.02 + 9\alpha) = 1$ or $0.1 + 45\alpha = 1$, so $\alpha = \mathbf{0.02}$ | A1 | $\alpha = 0.02$ must be exact and come from correct working |
## Part (e):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(F=5 \mid \text{Thomas' model}) = \mathbf{0.09}$ | B1ft | For value resulting from $0.01 + 4\times\text{"their }\alpha\text{"}$. If answer same as (c) or rounded version of (c), score B0 |
## Part (f):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Peta's model assumes the probability of hitting target is constant and Thomas' model assumes this probability increases with each attempt | B1 | Suitable comment about the probability of hitting the target. Allow idea that Peta's model suggests the dart may never hit the target but Thomas' says it will hit at least once (in first 10 throws) |
---\begin{enumerate}
\item In an experiment a group of children each repeatedly throw a dart at a target. For each child, the random variable $H$ represents the number of times the dart hits the target in the first 10 throws.
\end{enumerate}
Peta models $H$ as $\mathrm { B } ( 10,0.1 )$\\
(a) State two assumptions Peta needs to make to use her model.\\
(b) Using Peta's model, find $\mathrm { P } ( H \geqslant 4 )$
For each child the random variable $F$ represents the number of the throw on which the dart first hits the target.
Using Peta's assumptions about this experiment,\\
(c) find $\mathrm { P } ( F = 5 )$
Thomas assumes that in this experiment no child will need more than 10 throws for the dart to hit the target for the first time. He models $\mathrm { P } ( F = n )$ as
$$\mathrm { P } ( F = n ) = 0.01 + ( n - 1 ) \times \alpha$$
where $\alpha$ is a constant.\\
(d) Find the value of $\alpha$\\
(e) Using Thomas' model, find $\mathrm { P } ( F = 5 )$\\
(f) Explain how Peta's and Thomas' models differ in describing the probability that a dart hits the target in this experiment.
\hfill \mbox{\textit{Edexcel Paper 3 2018 Q3 [11]}}