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### MATHEMATICS 9709

#### Cambridge International AS and A Level

 Name of student Date Adm. number Year/grade Stream Subject Probability & Statistics 1 (S1) Variant(s) P63 Start time Duration Stop time

Qtn No. 1 2 3 4 5 6 7 Total
Marks 5 5 6 5 6 9 9 45
Score

Get Mathematics 9709 Topical Questions (2010-2021) $14.5 per Subject. Attempt all the 7 questions Question 1 Code: 9709/63/M/J/14/1, Topic: Representation of data Some adults and some children each tried to estimate, without using a watch, the number of seconds that had elapsed in a fixed time-interval. Their estimates are shown below. $$\begin{array}{llllllllllllllll} \text {Adults:} & 55 & 58 & 67 & 74 & 63 & 61 & 63 & 71 & 56 & 53 & 54 & 78 & 73 & 64 & 62 \\ \text {Children:} & 86 & 95 & 89 & 72 & 61 & 84 & 77 & 92 & 81 & 54 & 43 & 68 & 62 & 67 & 83 \end{array}$$$\text{(i)}$Draw a back-to-back stem-and-leaf diagram to represent the data.$[3]\text{(ii)}$Make two comparisons between the estimates of the adults and the children.$[2]$Question 2 Code: 9709/63/M/J/12/2, Topic: Representation of data The heights,$x \mathrm{~cm}$, of a group of young children are summarised by $$\Sigma(x-100)=72, \quad \Sigma(x-100)^{2}=499.2$$ The mean height is$104.8 \mathrm{~cm}$.$\text{(i)}$Find the number of children in the group.$\text{(ii)}$Find$\Sigma(x-104.8)^{2}$. Question 3 Code: 9709/63/M/J/14/2, Topic: The normal distribution There is a probability of$\frac{1}{7}$that Wenjie goes out with her friends on any particular day. 252 days are chosen at random.$\text{(i)}$Use a normal approximation to find the probability that the number of days on which Wenjie goes out with her friends is less than than 30 or more than 44.$[5]\text{(ii)}$Give a reason why the use of a normal approximation is justified.$[1]$Question 4 Code: 9709/63/M/J/15/2, Topic: Probability When Joanna cooks, the probability that the meal is served on time is$\frac{1}{5}$. The probability that the kitchen is left in a mess is$\frac{3}{5}$. The probability that the meal is not served on time and the kitchen is not left in a mess is$\frac{3}{10}$. Some of this information is shown in the following table. $$\begin{array}{|l|c|c|c|} \hline & \text{Kitchen left in a mess} & \text{Kitchen not left in a mess} & \text{Total} \\ \hline \text{Meal served on time} & & & \frac{1}{5} \\ \hline \text{Meal not served on time} & & \frac{3}{10} & \\ \hline \text{Total} & & & 1 \\ \hline \end{array}$$$\text{(i)}$Copy and complete the table.$[3]\text{(ii)}$Given that the kitchen is left in a mess, find the probability that the meal is not served on time.$[2]$Question 5 Code: 9709/63/M/J/18/2, Topic: The normal distribution The random variable$X$has the distribution$\mathrm{N}\left(-3, \sigma^{2}\right)$. The probability that a randomly chosen value of$X$is positive is$0.25$.$\text{(i)}$Find the value of$\sigma$.$[3]\text{(ii)}$Find the probability that, of 8 random values of$X$, fewer than 2 will be positive.$[3]$Question 6 Code: 9709/63/M/J/12/3, Topic: Permutations and combinations$\text{(i)}$In how many ways can all 9 letters of the word TELEPHONE be arranged in a line if the letters$\mathrm{P}$and$\mathrm{L}$must be at the ends?$[1]$How many different selections of 4 letters can be made from the 9 letters of the word TELEPHONE if$\text{(ii)}$there are no Es,$[1]\text{(iii)}$there is exactly$1 \mathrm{E}$,$[2]\text{(iv)}$there are no restrictions?$[4]$Question 7 Code: 9709/63/M/J/20/5, Topic: Discrete random variables A pair of fair coins is thrown repeatedly until a pair of tails is obtained. The random variable$X$denotes the number of throws required to obtain a pair of tails.$\text{(a)}$Find the expected value of$X$.$[1]\text{(b)}$Find the probability that exactly 3 throws are required to obtain a pair of tails.$[1]\text{(c)}$Find the probability that fewer than 6 throws are required to obtain a pair of tails.$[2]$On a different occasion, a pair of fair coins is thrown 80 times.$\text{(d)}$Use an approximation to find the probability that a pair of tails is obtained more than 25 times.$[5]\$

Worked solutions: P1, P3 & P6 (S1)

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