$\def\frac{\dfrac}$
43(a) $k=0.25$ (b) $E(x)=1.95$
44(a) $\quad 6 k^{3}$ (b) $k=0.3$ (c) $0.4$
45(a) $\quad p=\frac{1}{10}$ (b) $\mathrm{E}(x)=\frac{11}{10}$
46(a) $\frac{3}{8}$ (b) (i) $p=\frac{4}{8}, q=\frac{5}{8}$ (or) $p=\frac{5}{8}, q=\frac{4}{8}$ (ii) $r=\frac{5}{16}, k=9$ (c) $\frac{6}{11}$
47(a) $\frac{3}{8}$ (b) $\frac{2}{7}, \frac{5}{7}, \frac{3}{7}$ (c) $\frac{5}{14}$
48 $k=25$
49(a) $0.0478$ (b) $\sigma=4.87$ (c) (i) $0.968$ (ii) $0.988$ (d) $0.786$
50(a) $0.819$ (b) $\sigma=200$ (c) $t=48.7$ (d) (i) $0.360$ (ii) $0.924$
51(a) $60$ (b) (i) $c=87$ (iii) $d=52$
52(a) $\sigma=3.65$ (b) $w=11,1$ (c) $\frac{57}{95}$ (d) 200
53(a) 3 (b)(i) $p=30$ (ii) $q=60$
1  (IB/sl/2019/November/Paper2/q7)
[Maximum mark: 7 ] The following table shows the probability distribution of a discrete random variable X, where $a \geq 0$ and $b \geq 0$ $$\begin{array}{ccccc}\hline x & 1 & 4 & a & a+b0.5 \\\hline \mathrm{P}(X=x) & 0.2 & 0.5 & b & a \\\hline\end{array} $$

2  (IB/sl/2019/May/paper1tz2/ql)
[Maximum mark: 6] The following table shows the probability distribution of a discrete random variable $X$. $$\begin{array}{ccccc}\hline X & 0 & 1 & 2 & 3 \\\hline \mathrm{P}(X=x) & \frac{3}{13} & \frac{1}{13} & \frac{4}{13} & k \\\hline\end{array}$$

3  (IB/sl/2019/May/paper2tz1/q1)
[Maximum mark: 5] Ten students were asked for the distance, in $\mathrm{km}$, from their home to school. Their responses are recorded below. $\begin{array}{llllllllll}0.3 & 0.4 & 3 & 3 & 3.5 & 5 & 7 & 8 & 8 & 10\end{array}$
The following boxandwhisker plot represents this data.

4  (IB/s1/2019/May/paper2tz1/q10)
[Maximum mark: 16] There are three fair sixsided dice. Each die has two green faces, two yellow faces and two red faces. All three dice are rolled.
Ted plays a game using these dice. The rules are: Having a turn means to roll all three dice. He wins $\$ 10$ for each green face rolled and adds this to his winnings. After a turn Ted can either end the game (and keep his winnings), or have another turn (and try to increase his winnings). If two or more red faces are rolled in a turn, all winnings are lost and the game ends.
The random variable $D(\mathrm{~S})$ represents how much is added to his winnings after a turn. The following table shows the distribution for $D$, where $\$ w$ represents his winnings in the game so far. $$\begin{array}{cccccc}\hline D(\$)& w & 0 & 10 & 20 & 30 \\\hline \mathrm{P}(D=d) & x & y & \frac{1}{3} & \frac{2}{9} & \frac{1}{27} \\\hline\end{array}$$
Ted will always have another turn if he expects an increase to his winnings.

5  (IB/sl/2018/November/Paper1/q9)
[Maximum mark: 15] A bag contains $n$ marbles, two of which are blue. Hayley plays a game in which she randomly draws marbles out of the bag, one after another, without replacement. The game ends when Hayley draws a blue marble.
Hayley plays the game when $n=5 .$ She pays $\$ 20$ to play and can earn money back depending on the number of draws it takes to obtain a blue marble. She earns no money back if she obtains a blue marble on her first draw. Let $M$ be the amount of money that she earns back playing the game. This information is shown in the following table,$$\begin{array}{ccccc}\hline \text{Number of draws} & 1 & 2 & 3 & 4 \\\hline \text{Money earned back} \mathbf{( S} M ) & 0 & 20 & 8 k & 12 k \\\hline\end{array}$$

6  (IB/sl/2018/May/paper2tz1/q2)
[Maximum mark: 6] A biased foursided die is rolled. The following table gives the probability of each score. $$\begin{array}{ccccc}\hline \text{Score }& 1 & 2 & 3 & 4 \\\hline \text{Probability} & 0.28& k & 0.15 & 0.3 \\\hline\end{array}$$

7  $(\mathrm{IB} / \mathrm{sl} / 2017 /$ November $/$ Paper $2 / \mathrm{q} 4)$
[Maximum mark: 8] A discrete random variable $X$ has the following probability distribution. $$\begin{array}{ccccc}\hline X & 0 & 1 & 2 & 3 \\\hline \mathrm{P}(X=x) & 0.475 & 2 h^{2} & \frac{k}{10} & 6 k^{2} \\\hline\end{array}$$

8  (IB/s1/2017/May/paper1tz1/q10)
[Maximum mark: 15] The following table shows the probability distribution of a discrete random variable $A$, in terms of an angle $\theta$, $$\begin{array}{ccc}\hline a & 1 & 2 \\\hline \mathrm{P}(A=a) & \cos \theta & 2 \cos 2 \theta \\\hline\end{array}$$

9  (IB/s1/2017/May/paper2tz2/q10)
[Maximum mark: 15] The following table shows a probability distribution for the random variable $X$, where $\mathrm{E}(X)=1.2$, $$\begin{array}{ccccc}\hline x & 0 & 1 & 2 & 3 \\\hline \mathrm{P}(X=x) & p & \frac{1}{2} & \frac{3}{10} & q \\\hline\end{array}$$
A bag contains white and blue marbles, with at least three of each colour. Three marbles are drawn from the bag. without replacement. The number of blue marbles drawn is given by the random variable $X$.
A game is played in which three marbles are drawn from the bag of ten marbles, without replacement. A player wins a prize if three white marbles are drawn.

10  (IB/s1/2016/May/paper2tz1/q8)
[Maximum mark: 15] A factory has two machines, $A$ and $B$. The number of breakdowns of each machine is independent from day to day. Let $A$ be the number of breakdowns of Machine A on any given day. The probability distribution for $A$ can be modelled by the following table. $$\begin{array}{cllll}\hline a & 0 & 1 & 2 & 3 \\\hline \mathrm{P}(A=a) & 0.55 & 0.3 & 0.1& k$\\\hline\end{array}$$
Let $B$ be the number of breakdowns of Machine $\mathrm{B}$ on any given day. The probability distribution for $B$ can be modelled by the following table.$$\begin{array}{cllll}\hline b & 0 & 1 & 2 & 3 \\\hline \mathrm{P}(B=b) & 0.7 & 0.2 & 0.08 & 0.02 \\\hline\end{array}$$
On Tuesday, the factory uses both Machine $\mathrm{A}$ and Machine $\mathrm{B}$. The variables $A$ and $B$ are independent.

11  (IB/s1/2019/November/Paper1/q7)
[Maximum mark: 6] Let $X$ and $Y$ be normally distributed with $X \sim \mathrm{N}\left(14, a^{2}\right)$ and $Y \sim \mathrm{N}\left(22, a^{2}\right), a>0$.
It is given that $\mathrm{P}(X>20)=0.112$.

12  (IB/s1/2019/November/Paper2/q9)
[Maximum mark: 15] SpeedWay airline flies from city $\mathrm{A}$ to city $\mathrm{B}$. The flight time is normally distributed with a mean of 260 minutes and a standard deviation of 15 minutes. A flight is considered late if it takes longer than 275 minutes.
The flight is considered to be on time if it takes between $m$ and 275 minutes, The probability that a flight is on time is $0.830$.
During a week, SpeedWay has 12 flights from city A to city B. The time taken for any flight is independent of the time taken by any other flight.
SpeedWay increases the number of flights from city A to city B to 20 flights each week, and improves their efficiency so that more flights are on time. The probability that at least 19 flights are on time is $0.788$.

13  (IB/s1/2019/May/paper1tz1/q9)
[Maximum mark: 13] A random variable $Z$ is normally distributed with mean 0 and standard deviation 1 . It is known that $\mathrm{P}(z < 1.6)=a$ and $\mathrm{P}(z>2.4)=b$. This is shown in the following diagram.
A second random variable $X$ is normally distributed with mean $m$ and standard deviation $s$. It is known that $\mathrm{P}(x < 1)=a$.
It is also known that $\mathrm{P}(x>2)=b$.

14  (IB/s1/2019/May/paper2tz2/q9)
[Maximum mark: 14] At Penna Airport the probability, $\mathrm{P}(A)$, that all passengers arrive on time for a flight is $0.70$, The probability, $\mathrm{P}(D)$, that a flight departs on time is $0.85$. The probability that all passengers arrive on time for a fight and it departs on time is $0.65$.
The number of hours that pilots fly per week is normally distributed with a mean of 25 hours and a standard deviation $\sigma .90 \%$ of pilots fly less than 28 hours in a week.

15  (IB/s1/2018/November/Paper2/q9)
[Maximum mark: 15] A nationwide study on reaction time is conducted on participants in two age groups. The participants in Group $X$ are less than 40 years old. Their reaction times are normally distributed with mean $0.489$ seconds and standard deviation $0.07$ seconds.
The participants in Group $Y$ are 40 years or older. Their reaction times are normally distributed with mean $0.592$ seconds and standard deviation $\sigma$ seconds.
In the study, $38 \%$ of the participants are in Group $X$.
Find the probability that the participant is in Group $\mathrm{X}$.
Find the probability that at least two of them are in Group $X$. 
16  (IB/s1/2018/May/paper2tz1/q9)
[Maximum mark: 17] The weights, in grams, of oranges grown in an orchard, are normally distributed with a mean of $297 \mathrm{~g}$. It is known that $79 \%$ of the oranges weigh more than $289 \mathrm{~g}$ and $9.5 \%$ of the oranges weigh more than $310 \mathrm{~g}$.
The weights of the oranges have a standard deviation of $\sigma$,
The grocer at a local grocery store will buy the oranges whose weights exceed the 35 th percentile.
The orchard packs oranges in boxes of 36 .
The grocer selects two boxes at random.

17  (IB/s1/2018/May/paper2tz2/q10)
[Maximum mark: 17] The mass $M$ of apples in grams is normally distributed with mean $\mu$. The following table shows probabilities for values of $M$. $$\begin{array}{cccc}\hline Values of M & M < 93 & 93 \leq M \leq 119 & M> 119 \\\hline \mathbf{P}(\boldsymbol{X}) & k & 0.98 & 0.01 \\\hline\end{array}$$
The apples are packed in bags of ten. Any apples with a mass less than $95 \mathrm{~g}$ are classified as small.

18  (IB/s1/2017/November/Paper2/q7)
[Maximum mark: 7] The heights of adult males in a country are normally distributed with a mean of $180 \mathrm{~cm}$ and a standard deviation of $\sigma \mathrm{cm} .17 \%$ of these men are shorter than $168 \mathrm{~cm} .80 \%$ of them have heights between $(192h) \mathrm{cm}$ and $192 \mathrm{~cm}$. Find the value of $h$. 
19  (IB/s1/2017/May/paper1tz2/q3)
[Maximum mark: 6] The random variable $X$ is normally distributed with a mean of 100 . The following diagram shows the normal curve for $X$. Let $R$ be the shaded region under the curve, to the right of 107 . The area of $R$ is $0.24$.

20  (IB/s1/2017/May/paper2tz1/q9)
[Maximum mark: 15] A random variable $X$ is normally distributed with mean, $\mu$. In the following diagram, the shaded region between 9 and $\mu$ represents $30 \%$ of the distribution.
The standard deviation of $X$ is $2.1$.
The random variable $Y$ is normally distributed with mean $\lambda$ and standard deviation $3.5$, The events $X>9$ and $Y>9$ are independent, and $P((X>9) \cap(Y>9))=0.4$.

21  (IB/s1/2016/May/paper2tz1/q1)
[Maximum mark: 6] A random variable $X$ is distributed normally with a mean of 20 and standard deviation of 4 .

22  (IB/s1/2016/May/paper2tz2/q6)
[Maximum mark: 6] A competition consists of two independent events, shooting at 100 targets and running for one hour. The number of targets a contestant hits is the $S$ score. The $S$ scores are normally distributed with mean 65 and standard deviation 10 .
The distance in $\mathrm{km}$ that a contestant runs in one hour is the $R$ score. The $R$ scores are normally distributed with mean 12 and standard deviation $2.5$. The $R$ score is independent of the $S$ score. Contestants are disqualified if their $S$ score is less than 50 and their $R$ score is less than $x \mathrm{~km}$.

23  (IB/sl/2019/May/paper1tz1/q1)
[Maximum mark: 6] The following Venn diagram shows the events $A$ and $B$, where $\mathrm{P}(A)=0.3$. The values shown are probabilities.

24  (IB/s1/2018/May/paper2tz1/q5)
[Maximum mark: 6] Two events $A$ and $B$ are such that $\mathrm{P}(A)=0.62$ and $\mathrm{P}(A \cap B)=0.18$.

25  (IB/s1/2016/November/Paper1/q5)
[Maximum mark: 6] Events $A$ and $B$ are independent with $\mathrm{P}(A \cap B)=0.2$ and $\mathrm{P}\left(A^{\prime} \cap B\right)=0.6$.

26  (IB/sl/2018/May/paper1tz2/q8)
[Maximum mark: 14] Pablo drives to work. The probability that he leaves home before $07: 00$ is $\frac{3}{4}$. If he leaves home before $07: 00$ the probability he will be late for work is $\frac{1}{8}$. If he leaves home at $07: 00$ or later the probability he will be late for work is $\frac{5}{8}$.

28  (IB/s1/2017/May/paper2tz1/g4)
[Maximum mark: 6] In a large university the probability that a student is left handed is $0.08$. A sample of 150 students is randomly selected from the university. Let $k$ be the expected number of lefthanded students in this sample.

29  (IB/s1/2016/November/Paper2/q5)
[Maximum mark: 6] The weights, $W$, of newborn babies in Australia are normally distributed with a mean $3.41 \mathrm{~kg}$ and standard deviation $0.57 \mathrm{~kg}$. A newborn baby has a low birth weight if it weighs less than $w \mathrm{~kg}$.
Find the probability that the baby weighs at least $2.15 \mathrm{~kg}$. 
30  (IB/sl/2016/November/Paper2/q7)
[Maximum mark; 6] A jar contains 5 red discs, 10 blue discs and $m$ green discs. A disc is selected at random and replaced. This process is performed four times.

31  (IB/s1/2016/May/paper1tz1/q2)
[Maximum mark: 6] The following Venn diagram shows the events $A$ and $B$, where $\mathrm{P}(A)=0.4, \mathrm{P}(A \cup B)=0.8$ and $P(A \cap B)=0,1$. The values $p$ and $q$ are probabilities.

32  (IB/s1/2016/May/paper1tz2/q8)
[Maximum mark: 13] In a class of 21 students, 12 own a laptop, 10 own a tablet, and 3 own neither. The following Venn diagram shows the events "own a laptop" and "own a tablet". The values $p, q, r$ and $s$ represent numbers of students.

33  (IB/s1/2019/May/paper1tz2/q8)
[Maximum mark: 16] A group of 10 girls recorded the number of hours they spent watching television during a particular week. Their results are summarized in the boxandwhisker plot below.
The group of girls watched a total of 180 hours of television.
A group of 20 boys also recorded the number of hours they spent watching television that same week. Their results are summarized in the table below. $$\begin{array}{ll}\hline \bar{x}=21 & \sigma=3\\\hline\end{array}$$
The following week, the group of boys had exams. During this exam week, the boys spent half as much time watching television compared to the previous week.

34  (IB/s1/2018/May/paper1tz1/q2)
[Maximum mark: 6] The following boxandwhisker plot shows the number of text messages sent by students in a school on a particular day.

35  (IB/sl/2018/May/paper1tz2/q3)
[Maximum mark: 6] A data set has $n$ items. The sum of the items is 800 and the mean is 20 .
The standard deviation of this data set is 3 . Each value in the set is multiplied by 10 .

36  (IB/s1/2017/May/paper1tz2/q8)
[Maximum mark: 14] A city hired 160 employees to work at a festival. The following cumulative frequency curve shows the number of hours employees worked during the festival.
The city paid each of the employees $£ 8$ per hour for the first 40 hours worked, and $£ 10$ per hour for each hour they worked after the first 40 hours.

37  (IB/s1/2017/May/paper2tz1/q1)
[Maximum mark: 7] Consider the following frequency table.$$\begin{array}{cc}\hline x & \text{Frequency} \\\hline 2 & 8 \\\hline 4 & 15 \\\hline 7 & 21 \\\hline 10 & 28 \\\hline 11 & 3 \\\hline\end{array}$$

38  (IB/sl/2016/November/Paper2/q8)
[Maximum mark: 16] Ten students were surveyed about the number of hours, $x$, they spent browsing the Internet during week 1 of the school year. The resuits of the survey are given below. $$\sum_{i=1}^{10} x_{i}=252, \sigma=5 \text { and median }=27$$

39  Question (IB/sl/2016/November/Paper2/q8b) 38 continued 
40  (IB/sl/2016/May/paper1tz1/q8)
[Maximum mark: 15] A school collects cans for recycling to raise money. Sam's class has 20 students. The number of cans collected by each student in Sam's class is shown in the following stem and leaf diagram. $$\begin{array}{rl} \text{Stem }& \text{Leaf }\\\hline 2 & 0,1,4,9,9 \\3 & 1,7,7,7,8,8 \\4 & 1,2,2,3,5,6,7,8 \\5 & 0\end{array}$$ Key: $3 \mid 1$ represents 31 cans
The following boxandwhisker plot also displays the number of cans collected by students in Sam's class.
There are 80 students in the school.

41  Question (IB/sl/2016/May/paper1tz1/q8b) 40 continued

42  (IB/s1/2016/May/paper1tz2/q2)
[Maximum mark: 5] There are 10 items in a data set. The sum of the items is 60 .
The variance of this data set is $3 .$ Each value in the set is multiplied by 4 .

43  (IB/sl/2015/November/Paper2/q2)
[Maximum mark: 5] The following table shows the probability distribution of a discrete random variable $X$. $$\begin{array}{ccccc}\hline x & 0 & 1 & 2 & 3 \\ \hline \mathrm{P}(X=x) & 0.15 & k & 0.1 & 2 k \\ \hline\end{array}$$

44  (IB/s1/2015/November/Paper2/q5)
[Maximum mark: 7] Let $C$ and $D$ be independent events, with $\mathrm{P}(C)=2 k$ and $\mathrm{P}(D)=3 k^{2}$, where $ 0 < k < 0.5$.

45  (IB/s1/2015/May/paper1tz1/q1)
[Maximum mark: 6] A discrete random variable $X$ has the following probability distribution. $$\begin{array}{ccccc}\hline x & 0 & 1 & 2 & 3 \\ \hline P(X=x) & \frac{3}{10} & \frac{4}{10} & \frac{2}{10} & p \\ \hline\end{array}$$

46  $\langle\mathrm{IB} / \mathrm{sl} / 2015 /$ May $/$ paper $1 \mathrm{tz} 1 / \mathrm{q} 10)$
[Maximum mark: 15] Ann and Bob play a game where they each have an eightsided die. Ann's die has three green faces and five red faces; Bob's die has four green faces and four red faces. They take turns rolling their own die and note what colour faces up. The first player to roll green wins, Ann rolls first. Part of a tree diagram of the game is shown below.
Find the value of $r$ and of $k$.

47  (IB/s1/2015/May/paper1tz2/q1)
[Maximum mark: 6] A bag contains eight marbles. Three marbles are red and five are blue. Two marbles are drawn from the bag without replacement.

48  (IB/s1/2015/May/paper1tz2/q7)
[Maximum mark: 7] A bag contains black and white chips. Rose pays $\$ 10$ to play a game where she draws a chip from the bag. The following table gives the probability of choosing each colour chip. $$\begin{array}{lcc}\hline \text{Outcome} & \text{black} & \text{white }\\\hline \text{Probability} & 0.4 & 0.6 \\\hline\end{array}$$ Rose gets no money if she draws a white chip, and gets $S k$ if she draws a black chip. The game is fair. Find the value of $k$. 
49  (IB/s1/2015/May/paper2tz1/q9)
[Maximum mark: 16] A company makes containers of yogurt. The volume of yogurt in the containers is normally distributed with a mean of $260 \mathrm{ml}$ and standard deviation of $6 \mathrm{ml}$. A container which contains less than $250 \mathrm{ml}$ of yogurt is underfilled.
The company decides that the probability of a container being underfilled should be reduced to $0.02$. It decreases the standard deviation to $\sigma$ and leaves the mean unchanged.
The company changes to the new standard deviation, $\sigma$, and leaves the mean unchanged. A container is chosen at random for inspection. It passes inspection if its volume of yogurt is between 250 and $271 \mathrm{ml}$.

50  (IB/s1/2015/May/paper2tz2/q9)
[Maximum mark: 16] A machine manufactures a large number of nails. The length, $L \mathrm{~mm}$, of a nail is normally distributed, where $L\mathrm{N}\left(50, \sigma^{2}\right)$.
Show that $\sigma2.00$ (correct to three significant figures). All nails with length at least $t \mathrm{~mm}$ are classified as large nails.

51  (IB/s1/2015/November/Paper1/q1)
[Maximum mark: 5] The following boxandwhisker plot represents the examination scores of a group of students.
The range of the scores is 47 marks, and the interquartile range is 22 marks.

52  (IB/s1/2015/November/Paper2/q10)
[Maximum mark: 14] The masses of watermelons grown on a farm are normally distributed with a mean of $10 \mathrm{~kg}$. The watermelons are classified as small, medium or large. A watermelon is $s m a l l$ if its mass is less than $4 \mathrm{~kg}$. Five percent of the watermelons are classified as small.
The following table shows the percentages of small, medium and large watermelons grown on the farm. $$\begin{array}{ccc}\hline \text{small} & \text{medium} & \text{large }\\\hline 5 \% & 57 \% & 38 \% \\\hline\end{array}$$ A watermelon is large if its mass is greater than $w \mathrm{~kg}$.
All the medium and large watermelons are delivered to a grocer.
The grocer sells all the medium watermelons for $\$ 1.75$ each, and all the large watermelons for $\$ 3.00$ each. His costs on this delivery are $\$ 300$, and his total profit is, $\$ 150$. Find the number of watermelons in the delivery. 
53  (IB/s1/2015/May/paper1tz2/q3)
[Maximum mark: 6] The following cumulative frequency diagram shows the lengths of 160 fish, in $\mathrm{cm}$.
The following frequency table also gives the lengths of the 160 fish. $$\begin{array}{lcccc}\hline \text{Length }x \mathrm{~cm} & 0 \leq x \leq 2 & 2 < x \leq 3 & 3 < x \leq 4.5 & 4.5 < x \leq 6 \\\hline \text{Frequency} & p & 50 & q & 20 \\\hline\end{array}$$

Answer
1(a) Show (b) .0625
2(a) $k=\frac{5}{13}$ (b) $\dfrac{24}{13}$
3(a) 4.82 (b) 4.25 (c) 5
40(a)(i) $\frac 49$ (ii) $\frac{7}{27}$ (b) $\frac 13$ (c)(i)$\frac{7}{27}$ (ii) $\frac{4}{27}$ (d) 40
5(a)(i) $\frac{2}{n}$ (ii) $\frac{2(n2)}{n(n1)}$ (b) (i) $\frac{1}{5}$ (ii) $\frac{1}{10}$ (c) $k=5$
6 (a) $k=0.27$ (b) $E(x)=2.47$ (c) 12
7(a) $k=0.25$ (b) $P(x=2)=0.025$ (c) $P(x=2 \mid x>0)=\frac{1}{21}$
8 (a) Show (b) $\tan \theta=\frac{\sqrt{7}}{3}$ (c) $V=\pi\frac{\pi \sqrt{7}}{3}$
9(a)(i) $q=\frac{1}{30}$ (ii) $p=\frac{1}{6}$ (b) (i) $\frac{1}{30}$ (ii) $\frac{1}{6}$ (iii) $w=6$ (c) $0.279$ (d) $0.0651$
10 (a) $k=0.05$ (b) (i) $0.55$ (ii) $0.206$ (c) $0.42$ (d) (i) $0.174$ (ii) $0.402$
11(a) 18 (b) 0.776
12(a) .841 (b) 226 (c)(i) .991 (ii) .299 (d) .957
13(a) $1ab$ (b) $\frac{1ab}{1a}$ (c) $1.6$ (d) $\frac{1}{4}$
14(a) Show (b) (i) $0.05$ (ii) $\frac{1}{14}$ (c) $\sigma=2.34$ (d) $0.0267 \%$
15(a) $0.0107$ (b) $\sigma=0.220$ (c) $0.0163270$ (d) $0.0110$
16(a) $\quad P(289<w<310)=0.695$ (b) (i) $z=0.806$ (ii) $\sigma=9.92$ (c) 294 (d) $P(x>18)=0.954$ (e) $0.911$
17(a)(i) $\quad k=0.01$ (ii) $\mu=106$ (b) $P(x<95)=0.0245$ (c). 976 (d) (i) $48.8$ (ii) $0.885$
18 $ h=35.7$
19(a) $P(x>107)=0.24$ (b) $P(100<x<107)=0.26$ (c) $P(93<x<107)=0.52$
20(a) $P(x<9)=0.2$ (b) $\mu=10.8$ (c) $P(x>9)=0.8$ (d) $0.747$
21(a) Graph (b) $P(x \leqslant 25)=0.89$ (c) $c=22.1$
22 (a) $P(s<50)=0.0668$ (b) $x=9.41$
23(a) 0.2 (b) 0.3 (c) 0.9
24 (a) $P\left(A \cap B^{\prime}\right)=0.44$ (b) $P\left(A \mid B^{\prime}\right)=\frac{44}{63}$
25 (a) $0.8$ (b) $0.85$
26(a) $\frac{1}{8}, \frac{7}{8} ; \frac{1}{4}$ (b) $\frac{3}{32}$ (c) $\frac{1}{4}$ (d) $\frac{3}{8}$ (e) $\frac{7}{16}$
27(a) $\frac{3}{7}, \frac{5}{7}, \frac{2}{7}$ (b) $\frac{15}{28}$
28 (a) $k=12$ (b) (i) $0.119$ (11) $P(X<12)=0.457$
29 (a) $w=2.49$ (b) $0.745$
30 (a) $P($ red $)=\frac{5}{15+m}$ (b) $m=13$
31(a)(i) $\quad q=0.1$ (ii) $p=0.3$ (b) $\quad P(B)=0.5$
32(a)(i) $\quad p=3$ (ii) $r=8, s=6$ (b) (i) $\frac{4}{7}$ (ii) $\frac{2}{3}$ (c) (i) $\left[\right.$ First: $\left.\frac{4}{7}\right],\left[\right.$ Second: $\left.\frac{3}{5}, \frac{2}{5}\right]$ (ii) $\frac{11}{20}$
33(a) 24 (b) 5 (c) 18 (d) (i) 420 (ii) 20 (e)(i) 10.5 (ii) 2.25
34(a) $7$ (b) $k=22$
35 (a) $n=40$ (b) (i) 200 (ii) 900
36(a)(i) $40$ (ii) 130 (b) (i) 320 (ii) 350 (c) 18 (d) $k=520$
37(a)(i) $10$ (ii) 9 (b)(i) $7.15$ (ii) $\sigma^{2}=8.45$
38(a) $25.2$ (b) (i) $30.2$ (ii) $\sigma=5$ (c) (i) $25.7$ (ii) $22.6$ (d)(i) $70$ (ii) $k=35$
39
40(a) 38 (b) (i) $a=20$ (ii) $b=44$ (c) 55 (d) (i) 5 (ii) $20 \%$
41(e) (i) $41.4$ (ii) $18.5$
42(a) $6$ (b) (i) 24 (ii) 48
43(a) $k=0.25$ (b) $E(x)=1.95$
44(a) $\quad 6 k^{3}$ (b) $k=0.3$ (c) $0.4$
45(a) $\quad p=\frac{1}{10}$ (b) $\mathrm{E}(x)=\frac{11}{10}$
46(a) $\frac{3}{8}$ (b) (i) $p=\frac{4}{8}, q=\frac{5}{8}$ (or) $p=\frac{5}{8}, q=\frac{4}{8}$ (ii) $r=\frac{5}{16}, k=9$ (c) $\frac{6}{11}$
47(a) $\frac{3}{8}$ (b) $\frac{2}{7}, \frac{5}{7}, \frac{3}{7}$ (c) $\frac{5}{14}$
48 $k=25$
49(a) $0.0478$ (b) $\sigma=4.87$ (c) (i) $0.968$ (ii) $0.988$ (d) $0.786$
50(a) $0.819$ (b) $\sigma=200$ (c) $t=48.7$ (d) (i) $0.360$ (ii) $0.924$
51(a) $60$ (b) (i) $c=87$ (iii) $d=52$
52(a) $\sigma=3.65$ (b) $w=11,1$ (c) $\frac{57}{95}$ (d) 200
53(a) 3 (b)(i) $p=30$ (ii) $q=60$
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