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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 | 8 | 9 | 10 | 11 | 12 | Total |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Marks | 4 | 5 | 6 | 5 | 6 | 5 | 10 | 6 | 8 | 8 | 8 | 11 | 82 |

Score |

Question 1 Code: 9709/63/M/J/17/1, Topic: Probability

A biased die has faces numbered 1 to 6. The probabilities of the die landing on 1,3 or 5 are each equal to $0.1$. The probabilities of the die landing on 2 or 4 are each equal to $0.2$. The die is thrown twice. Find the probability that the sum of the numbers it lands on is 9. $[4]$

Question 2 Code: 9709/63/M/J/17/2, Topic: Discrete random variables

The probability that George goes swimming on any day is $\frac{1}{3}$. Use an approximation to calculate the probability that in 270 days George goes swimming at least 100 times. $[5]$

Question 3 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 4 Code: 9709/63/M/J/15/3, Topic: Discrete random variables

On a production line making cameras, the probability of a randomly chosen camera being substandard is $0.072$. A random sample of 300 cameras is checked. Find the probability that there are fewer than 18 cameras which are substandard. $[5]$

Question 5 Code: 9709/63/M/J/16/3, Topic: Discrete random variables

Two ordinary fair dice are thrown. The resulting score is found as follows.

If the two dice show different numbers, the score is the smaller of the two numbers.

If the two dice show equal numbers, the score is 0.

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

$\text{(ii)}$ Calculate the expected score. $[2]$

Question 6 Code: 9709/63/M/J/20/3, Topic: The normal distribution

In a certain town, the time, $X$ hours, for which people watch television in a week has a normal distribution with mean $15.8$ hours and standard deviation $4.2$ hours.

$\text{(a)}$ Find the probability that a randomly chosen person from this town watches television for less than 21 hours in a week. $[2]$

$\text{(b)}$ Find the value of $k$ such that $\mathrm{P}(X < k)=0.75$. $[3]$

Question 7 Code: 9709/63/M/J/12/4, Topic: Discrete random variables

The six faces of a fair die are numbered $1,1,1,2,3,3$. The score for a throw of the die, denoted by the random variable $W$, is the number on the top face after the die has landed.

$\text{(i)}$ Find the mean and standard deviation of $W$. $[3]$

$\text{(ii)}$ The die is thrown twice and the random variable $X$ is the sum of the two scores. Draw up a probability distribution table for $X$. $[4]$

$\text{(iii)}$ The die is thrown $n$ times. The random variable $Y$ is the number of times that the score is 3. Given that $\mathrm{E}(Y)=8$, find $\operatorname{Var}(Y)$. $[3]$

Question 8 Code: 9709/63/M/J/19/4, Topic: Permutations and combinations

$\text{(i)}$ Find the number of ways a committee of 6 people can be chosen from 8 men and 4 women if there must be at least twice as many men as there are women on the committee. $[3]$

$\text{(ii)}$ Find the number of ways a committee of 6 people can be chosen from 8 men and 4 women if 2 particular men refuse to be on the committee together. $[3]$

Question 9 Code: 9709/63/M/J/16/5, Topic: The normal distribution

The heights of school desks have a normal distribution with mean $69 \mathrm{~cm}$ and standard deviation $\sigma \mathrm{cm}$. It is known that $15.5 \%$ of these desks have a height greater than $70 \mathrm{~cm}$.

$\text{(i)}$ Find the value of $\sigma$. $[3]$

When Jodu sits at a desk, his knees are at a height of $58 \mathrm{~cm}$ above the floor. A desk is comfortable for Jodu if his knees are at least $9 \mathrm{~cm}$ below the top of the desk. Jodu's school has 300 desks.

$\text{(ii)}$ Calculate an estimate of the number of these desks that are comfortable for Jodu. $[5]$

Question 10 Code: 9709/63/M/J/17/5, Topic: Discrete random variables

Hebe attempts a crossword puzzle every day. The number of puzzles she completes in a week ( 7 days) is denoted by $X$.

$\text{(i)}$ State two conditions that are required for $X$ to have a binomial distribution. $[2]$

On average, Hebe completes 7 out of 10 of these puzzles.

$\text{(ii)}$ Use a binomial distribution to find the probability that Hebe completes at least 5 puzzles in a week. $[3]$

$\text{(iii)}$ Use a binomial distribution to find the probability that, over the next 10 weeks, Hebe completes 4 or fewer puzzles in exactly 3 of the 10 weeks. $[3]$

Question 11 Code: 9709/63/M/J/14/6, Topic: Discrete random variables

Tom and Ben play a game repeatedly. The probability that Tom wins any game is $0.3$. Each game is won by either Tom or Ben. Tom and Ben stop playing when one of them (to be called the champion) has won two games.

$\text{(i)}$ Find the probability that Ben becomes the champion after playing exactly 2 games. $[1]$

$\text{(ii)}$ Find the probability that Ben becomes the champion. $[3]$

$\text{(iii)}$ Given that Tom becomes the champion, find the probability that he won the 2nd game. $[4]$

Question 12 Code: 9709/63/M/J/15/7, Topic: Permutations and combinations

Rachel has 3 types of ornament. She has 6 different wooden animals, 4 different sea-shells and 3 different pottery ducks.

$\text{(i)}$ She lets her daughter Cherry choose 5 ornaments to play with. Cherry chooses at least 1 of each type of ornament. How many different selections can Cherry make? $[5]$

Rachel displays 10 of the 13 ornaments in a row on her window-sill. Find the number of different arrangements that are possible if

$\text{(ii)}$ she has a duck at each end of the row and no ducks anywhere else, $[3]$

$\text{(iii)}$ she has a duck at each end of the row and wooden animals and sea-shells are placed alternately in the positions in between. $[3]$