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Name of student | Date | ||||

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

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

Start time | Duration | Stop time |

Qtn No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Total |
---|---|---|---|---|---|---|---|---|

Marks | 7 | 6 | 8 | 10 | 9 | 12 | 10 | 62 |

Score |

Question 1 Code: 9709/61/O/N/15/4, Topic: Representation of data

$\text{(a)}$ Amy measured her pulse rate while resting, $x$ beats per minute, at the same time each day on 30 days. The results are summarised below.

$$ \Sigma(x-80)=-147 \quad \Sigma(x-80)^{2}=952 $$Find the mean and standard deviation of Amy's pulse rate. $[4]$

$\text{(b)}$ Amy's friend Marok measured her pulse rate every day after running for half an hour. Marok's pulse rate, in beats per minute, was found to have a mean of $148.6$ and a standard deviation of 18.5. Assuming that pulse rates have a normal distribution, find what proportion of Marok's pulse rates, after running for half an hour, were above 160 beats per minute. $[3]$

Question 2 Code: 9709/61/O/N/20/4, Topic: Discrete random variables

The random variable $X$ takes each of the values $1,2,3,4$ with probability $\frac{1}{4}$. Two independent values of $X$ are chosen at random. If the two values of $X$ are the same, the random variable $Y$ takes that value. Otherwise, the value of $Y$ is the larger value of $X$ minus the smaller value of $X$.

$\text{(a)}$ Draw up the probability distribution table for $Y$. $[4]$

$\text{(b)}$ Find the probability that $Y=2$ given that $Y$ is even. $[2]$

Question 3 Code: 9709/61/O/N/10/5, Topic: Probability

Three friends, Rick, Brenda and Ali, go to a football match but forget to say which entrance to the ground they will meet at. There are four entrances, $A, B, C$ and $D.$ Each friend chooses an entrance independently.

$\quad - $ The probability that Rick chooses entrance $A$ is $\frac{1}{3}$. The probabilities that he chooses entrances $B, C$ or $D$ are all equal.

$\quad - $ Brenda is equally likely to choose any of the four entrances.

$\quad - $ The probability that Ali chooses entrance $C$ is $\frac{2}{7}$ and the probability that he chooses entrance $D$ is $\frac{3}{5}$. The probabilities that he chooses the other two entrances are equal.

$\text{(i)}$ Find the probability that at least 2 friends will choose entrance $B$. $[4]$

$\text{(ii)}$ Find the probability that the three friends will all choose the same entrance. $[4]$

Question 4 Code: 9709/61/O/N/14/6, Topic: The normal distribution

A farmer finds that the weights of sheep on his farm have a normal distribution with mean $66.4 \mathrm{~kg}$ and standard deviation $5.6 \mathrm{~kg}$.

$\text{(i)}$ 250 sheep are chosen at random. Estimate the number of sheep which have a weight of between $70 \mathrm{~kg}$ and $72.5 \mathrm{~kg}$. $[5]$

$\text{(ii)}$ The proportion of sheep weighing less than $59.2 \mathrm{~kg}$ is equal to the proportion weighing more than $y \mathrm{~kg}$. Find the value of $y$. $[2]$

Another farmer finds that the weights of sheep on his farm have a normal distribution with mean $\mu \mathrm{kg}$ and standard deviation $4.92 \mathrm{~kg}. 25 \%$ of these sheep weigh more than $67.5 \mathrm{~kg}$.

$\text{(iii)}$ Find the value of $\mu$. $[3]$

Question 5 Code: 9709/61/O/N/16/6, Topic: Probability

Deeti has 3 red pens and 1 blue pen in her left pocket and 3 red pens and 1 blue pen in her right pocket. 'Operation $T$ ' consists of Deeti taking one pen at random from her left pocket and placing it in her right pocket, then taking one pen at random from her right pocket and placing it in her left pocket.

$\text{(i)}$ Find the probability that, when Deeti carries out operation $T$, she takes a blue pen from her left pocket and then a blue pen from her right pocket. $[2]$

The random variable $X$ is the number of blue pens in Deeti's left pocket after carrying out operation $T$.

$\text{(ii)}$ Find $\mathrm{P}(X=1)$. $[3]$

$\text{(iii)}$ Given that the pen taken from Deeti's right pocket is blue, find the probability that the pen taken from Deeti's left pocket is blue. $[4]$

Question 6 Code: 9709/61/O/N/12/7, Topic: Permutations and combinations

$\text{(a)}$ In a sweet shop 5 identical packets of toffees, 4 identical packets of fruit gums and 9 identical packets of chocolates are arranged in a line on a shelf. Find the number of different arrangements of the packets that are possible if the packets of chocolates are kept together. $[2]$

$\text{(b)}$ Jessica buys 8 different packets of biscuits. She then chooses 4 of these packets.

$\text{(i)}$ How many different choices are possible if the order in which Jessica chooses the 4 packets is taken into account? $[2]$

The 8 packets include 1 packet of chocolate biscuits and 1 packet of custard creams.

$\text{(ii)}$ How many different choices are possible if the order in which Jessica chooses the 4 packets is taken into account and the packet of chocolate biscuits and the packet of custard creams are both chosen? $[3]$

$\text{(c)}$ 9 different fruit pies are to be divided between 3 people so that each person gets an odd number of pies. Find the number of ways this can be done. $[5]$

Question 7 Code: 9709/61/O/N/19/7, Topic: The normal distribution

The shortest time recorded by an athlete in a $400 \mathrm{~m}$ race is called their personal best (PB). The PBs of the athletes in a large athletics club are normally distributed with mean $49.2$ seconds and standard deviation $2.8$ seconds.

$\text{(i)}$ Find the probability that a randomly chosen athlete from this club has a PB between 46 and 53 seconds. $[4]$

$\text{(ii)}$ It is found that $92 \%$ of athletes from this club have PBs of more than $t$ seconds. Find the value of $t$. $[3]$

Three athletes from the club are chosen at random.

$\text{(iii)}$ Find the probability that exactly 2 have PBs of less than 46 seconds. $[3]$