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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 |

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. $[5]$