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Name of student | HENRYTAIGO | Date | |||

Adm. number | Year/grade | HenryTaigo | Stream | HenryTaigo | |

Subject | Probability & Statistics 1 (S1) | Variant(s) | P61, P62, P63 | ||

Start time | Duration | Stop time |

Qtn No. | 1 | 2 | 3 | 4 | 5 | 6 | Total |
---|---|---|---|---|---|---|---|

Marks | 10 | 9 | 11 | 9 | 9 | 9 | 57 |

Score |

Question 1 Code: 9709/61/M/J/15/6, Topic: The normal distribution

$\text{(i)}$ In a certain country, $68 \%$ of households have a printer. Find the probability that, in a random sample of 8 households, 5,6 or 7 households have a printer. $[4]$

$\text{(ii)}$ Use an approximation to find the probability that, in a random sample of 500 households, more than 337 households have a printer. $[5]$

$\text{(iii)}$ Justify your use of the approximation in part $\text{(ii)}$. $[1]$

Question 2 Code: 9709/62/M/J/15/6, Topic: Permutations and combinations

$\text{(a)}$ Find the number of different ways the 7 letters of the word BANANAS can be arranged

$\text{(i)}$ if the first letter is $\mathrm{N}$ and the last letter is $\mathrm{B}$, $[3]$

$\text{(ii)}$ if all the letters $\mathrm{A}$ are next to each other. $[3]$

$\text{(b)}$ Find the number of ways of selecting a group of 9 people from 14 if two particular people cannot both be in the group together. $[3]$

Question 3 Code: 9709/63/M/J/15/6, Topic: Representation of data

Seventy samples of fertiliser were collected and the nitrogen content was measured for each sample. The cumulative frequency distribution is shown in the table below. $$ \begin{array}{|l|c|c|c|c|c|c|} \hline \text{Nitrogen content} & \leqslant 3.5 & \leqslant 3.8 & \leqslant 4.0 & \leqslant 4.2 & \leqslant 4.5 & \leqslant 4.8 \\ \hline \text{Cumulative frequency} & 0 & 6 & 18 & 41 & 62 & 70 \\ \hline \end{array} $$$\text{(i)}$ On graph paper draw a cumulative frequency graph to represent the data. $[3]$

$\text{(ii)}$ Estimate the percentage of samples with a nitrogen content greater than $4.4.$ $[2]$

$\text{(iii)}$ Estimate the median. $[1]$

$\text{(iv)}$ Construct the frequency table for these results and draw a histogram on graph paper. $[5]$

Question 4 Code: 9709/61/O/N/15/6, Topic: Probability

Nadia is very forgetful. Every time she logs in to her online bank she only has a $40 \%$ chance of remembering her password correctly. She is allowed 3 unsuccessful attempts on any one day and then the bank will not let her try again until the next day.

$\text{(i)}$ Draw a fully labelled tree diagram to illustrate this situation. $[2]$

$\text{(ii)}$ Let $X$ be the number of unsuccessful attempts Nadia makes on any day that she tries to $\log$ in to her bank. Copy and complete the following table to show the probability distribution of $X$. $[4]$

$$ \begin{array}{|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 \\ \hline \mathrm{P}(X=x) & & 0.24 & & \\ \hline \end{array} $$$\text{(iii)}$ Calculate the expected number of unsuccessful attempts made by Nadia on any day that she tries to $\log$ in. $[3]$

Question 5 Code: 9709/62/O/N/15/6, Topic: Discrete random variables

A fair spinner $A$ has edges numbered $1,2,3,3$. A fair spinner $B$ has edges numbered $-3,-2,-1,1$. Each spinner is spun. The number on the edge that the spinner comes to rest on is noted. Let $X$ be the sum of the numbers for the two spinners.

$\text{(i)}$ Copy and complete the table showing the possible values of $X$. $[1]$

$\text{(ii)}$ Draw up a table showing the probability distribution of $X$. $[3]$

$\text{(iii)}$ Find $\operatorname{Var}(X)$. $[3]$

$\text{(iv)}$ Find the probability that $X$ is even, given that $X$ is positive. $[2]$

Question 6 Code: 9709/63/O/N/15/6, Topic: Representation of data

The heights to the nearest metre of 134 office buildings in a certain city are summarised in the table below.

$$ \begin{array}{|l|c|c|c|c|c|} \hline \text{Height (m)} & 21-40 & 41-45 & 46-50 & 51-60 & 61-80 \\ \hline \text{Frequency} & 18 & 15 & 21 & 52 & 28 \\ \hline \end{array} $$$\text{(i)}$ Draw a histogram on graph paper to illustrate the data. $[4]$

$\text{(ii)}$ Calculate estimates of the mean and standard deviation of these heights. $[5]$