GAC010 Mathematics II: Probability, Statistics and Finance

A cheaper headline rate need not produce the cheaper total, and a larger survey need not produce a more representative result. Mathematics II uses explicit rules, calculations and evidence limits to examine such questions.

GAC010 covers sequences, finance, probability and statistics. Your centre's current brief determines assessment tasks, weights, tools and deadlines. A test 测验, projects 项目, an examination 考试 and coursework 平时作业 may use different instructions; these original practice sheets do not establish their official pattern or university credit.

Show the known data, why the formula applies, the symbolic equation and the substitution. State the unit and interpret the result. A statistical summary describes a defined set of observations or a stated model; it does not automatically establish a cause or a whole-population finding.

All financial amounts, rates, fees and pay rules in the examples are invented teaching conditions. Use their stated timing and rounding assumptions rather than assuming an actual tax or lending rule.

A sequence 数列 is an ordered list of numbers; a series 级数 is their sum.

An arithmetic sequence 等差数列 adds a fixed common difference 公差 $d$ each step. A geometric sequence 等比数列 multiplies by a fixed common ratio 公比 $r$.

• The sums are $S_n = \tfrac{n}{2}\left(2a + (n-1)d\right)$ and $S_n = \dfrac{a(1 - r^n)}{1 - r}$ for $r \neq 1$.
• A geometric series with $|r| < 1$ has a sum to infinity 无穷和 $S_\infty = \dfrac{a}{1-r}$.
• Keep a term separate from a total. Some financial models compound geometrically, while others use simple interest or cash flows. Match the model to the specified rule.

Worked example. row counts differ by a constant amount

Known: row counts are 4, 7 and 10, continuing with the same difference. Thus $a=4$ and $d=3$. For the fifth row:

For the first five rows together, use the arithmetic sum, not the fifth term.

Practice sheet 2.1 gives a progressively harder set with worked solutions.

This unit is the arithmetic of an ordinary life, in English.

• Gross pay 税前工资 is what you earn; net pay 税后工资 is what arrives after deductions 扣除项 such as tax and insurance.
• Simple interest 单利 is $I = Prt$ — interest on the original amount only.
• Compound interest is $A = P(1+r)^n$ — interest on the interest, which is why it is geometric.
• Depreciation 折旧 is a reduction in recorded value. A fixed-percentage model uses $V_n=V_0(1-r)^n$; other depreciation conventions can use different rules.
• A discount 折扣 of 20% multiplies by 0.8. Two successive discounts of 20% and 10% multiply by $0.8 \times 0.9 = 0.72$, which is a 28% reduction, not 30%.

Worked example. calculate regular and overtime hours separately

Known: 8 hours at 20 units per hour and 2 overtime hours at 1.5 times that rate. Separate the two categories before adding pay.

If the invented fixed deduction is 15 units, take it from gross pay once.

Practice sheet 2.2 gives a progressively harder set with worked solutions.

• Under a stated constant discount rate and timing assumption, the present value 现值 of a future amount is: $PV = \dfrac{FV}{(1+r)^n}$.
• An annuity 年金 has specified payments at regular intervals. A loan repayment 贷款还款 schedule may instead have changing amounts or an adjusted final payment; specify whether interest comes before or after payment.
• The label APR 年利率 is an annual rate measure whose calculation and included charges depend on the stated convention. This sheet uses explicitly defined period rates, not an unstated APR convention.
• ⚠ A nominal 12% teaching rate with monthly rate 1% has effective annual growth $(1.01)^{12}-1\approx12.68\%$. A different compounding frequency or fee schedule needs a different calculation.

Worked example. interest then payment, in that order

Known: opening balance 1000 units, period rate 2%, end-of-period payment 300 units. Interest comes before payment under this stated convention.

The next period's interest uses 720, not the original principal.

Practice sheet 2.3 gives a progressively harder set with worked solutions.

Probability 概率 measures how likely an event is, on a scale from 0 to 1.

• For equally likely outcomes, $P(A) = \dfrac{\text{favourable outcomes}}{\text{total outcomes}}$.
• Events are mutually exclusive 互斥 when they cannot both happen: $P(A \text{ or } B) = P(A) + P(B)$.
• Events are independent 独立 when one does not affect the other: $P(A \text{ and } B) = P(A) \times P(B)$.
• The complement 对立事件 rule, $P(\text{not } A) = 1 - P(A)$, is often the fastest route to an answer containing "at least one".
• Counting techniques 计数方法 supply the denominator: a permutation 排列 counts ordered selections, a combination 组合 counts unordered ones.

Conditional probability is $P(A\mid B)=P(A\cap B)/P(B)$ for $P(B)>0$. A union subtracts its overlap: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. Association in a supplied table does not by itself establish a cause.

Worked example. a second draw depends on the first

Known: two red and one blue counter, two random draws without replacement. To get two reds, follow the red-red path and multiply its conditional probabilities.

A first blue leaves only red counters; the blue-blue path has probability zero. The draws are not independent.

Practice sheet 2.4 gives a progressively harder set with worked solutions.

• A population 总体 is everyone you want to describe; a sample 样本 is who you actually measure.
• Generalising to a population needs a suitably representative 有代表性的 sample and an honest account of selection and nonresponse. A class survey can describe that class without establishing a school-wide percentage.
• Qualitative data 定性数据 are categories; quantitative data 定量数据 measure amounts, either discrete 离散 (counted) or continuous 连续 (measured).
• Bias 偏差 enters through who you ask, who answers, and how the question is worded, and arithmetic alone does not repair missing or systematically selected observations.

Worked example. equal bar heights can conceal unequal frequencies

Known: class 0 to 10 has frequency 20; class 10 to 30 has frequency 20. Widths differ, so calculate frequency density.

The second bar is half as high but twice as wide. Both areas represent 20 observations.

Practice sheet 2.5 gives a progressively harder set with worked solutions.

For unequal-width histogram classes, height is frequency divided by class width so that bar area represents frequency. Keep categories in separated bars and numeric continuous intervals in a histogram.

• The mean 平均数 is the total divided by the count. It uses every value, and every outlier.
• The median 中位数 is the middle value in order. Moving an extreme value without changing the ordered middle may leave it unchanged.
• The mode 众数 is the most frequent value, and it can identify the most frequent category without imposing a numerical scale. A distribution may have more than one mode or no unique mode.
• ⚠ Choose the summary for the question. A high extreme can pull the mean above the median. The mean still measures total per observation; the median locates the ordered middle. Neither says that most observations equal it.

Worked example. group size determines a combined mean

Known: group A has 10 scores with mean 60; group B has 20 with mean 75. Recover each group's total before combining.

The unweighted mean of 60 and 75 would be 67.5, which ignores the unequal counts.

Practice sheet 2.6 gives a progressively harder set with worked solutions.

• The range 全距 is the largest minus the smallest — the full span, so an extreme can change it substantially.
• The interquartile range (IQR) 四分位距 is the difference $Q_3-Q_1$ spanning the central half by the stated quartile convention. A box plot 箱线图 shows quartiles and a median, with whisker rules that must be specified.
• The standard deviation 标准差 summarises spread using squared deviations. For a complete population, use: $\sigma=\sqrt{\sum(x-\mu)^2/n}$. A sample estimate uses $s=\sqrt{\sum(x-\bar x)^2/(n-1)}$ under that stated convention. Variance has squared units; standard deviation has the original units.
• A larger standard deviation means a wider spread. Two data sets can share a mean and describe completely different situations.

Worked example. read a five-number summary rather than every observation

Known five-number summary: minimum 2, $Q_1=4$, median 6, $Q_3=8$, maximum 12. The overall span and middle-half span are different.

The box does not identify every original value, the sample size or why the spread arose.

Practice sheet 2.7 gives a progressively harder set with worked solutions.

For sheet 2.7, quartiles are medians of ordered half-lists, excluding the overall median for an odd count. Whiskers run to the minimum and maximum. Do not mix that convention with software interpolated quartiles or outlier-fence whiskers.

• The normal distribution 正态分布 is a continuous symmetric bell-shaped model specified by its mean and positive standard deviation. Those two summaries alone do not establish that observed data follow this model.
• Under a normal model, the approximate empirical rule is: about 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three.
• A z-score 标准分 says how many standard deviations a value sits from the mean: $z = \dfrac{x - \mu}{\sigma}$, which is what makes two different scales comparable.

For a normal model, use cumulative area $\Phi(z)=P(Z\leq z)$ and symmetry $\Phi(-z)=1-\Phi(z)$. An interval subtracts its two cumulative bounds. Expected count is sample size times probability and need not be an integer; an observed count is an integer.

Worked example. a threshold is an area beyond a standardised value

Known: an invented measurement model has $\mu=50$ and $\sigma=10$. For values above 60, standardise the threshold.

Use the complement of the supplied cumulative area.

This is a model probability, not a guarantee that exactly 15.87% of a finite sample exceeds 60.

Practice sheet 2.8 gives a progressively harder set with worked solutions.