| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2015 |
| Session | June |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared goodness of fit |
| Type | Chi-squared goodness of fit: Other continuous |
| Difficulty | Challenging +1.2 This is a multi-part Further Maths statistics question requiring chi-squared goodness of fit with geometric distribution, followed by a binomial probability calculation. While it involves multiple techniques and careful calculation (computing expected frequencies, combining cells, finding degrees of freedom, and nested binomial probabilities), the methods are standard applications without requiring novel insight. The geometric distribution setup is straightforward, and the final probability calculation, though involving composition of binomials, follows a clear template. Slightly above average difficulty due to the multi-step nature and Further Maths context. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities5.02f Geometric distribution: conditions5.06b Fit prescribed distribution: chi-squared test |
| \(x\) | 1 | 2 | 3 | 4 | 5 | 6 | \(\geqslant 7\) |
| Frequency | 126 | 43 | 22 | 3 | 5 | 1 | 0 |
Each of 200 identically biased dice is thrown repeatedly until an even number is obtained. The number of throws, $x$, needed is recorded and the results are summarised in the following table.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | }
\hline
$x$ & 1 & 2 & 3 & 4 & 5 & 6 & $\geqslant 7$ \\
\hline
Frequency & 126 & 43 & 22 & 3 & 5 & 1 & 0 \\
\hline
\end{tabular}
\end{center}
State a type of distribution that could be used to fit the data given in the table above.
Fit a distribution of this type in which the probability of throwing an even number for each die is 0.6 and carry out a goodness of fit test at the 5\% significance level.
For each of these dice, it is known that the probability of obtaining a 6 when it is thrown is 0.25 . Ten of these dice are each thrown 5 times. Find the probability that at least one 6 is obtained on exactly 4 of the 10 dice.
\hfill \mbox{\textit{CAIE FP2 2015 Q11 OR}}