GAC010 คณิตศาสตร์ II: ความน่าจะเป็น สถิติ และการเงิน
GAC คณิตศาสตร์ หัวข้อ 2 23:49 การบรรยายภาษาอังกฤษ · คำบรรยายภาษาอังกฤษ + 中文 ลอยตัวบนภาพ
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A cheaper headline rate need not produce the cheaper total, and a larger survey need not produce a more representative result. Mathematics two uses explicit rules, calculations and evidence limits to examine such questions. GAC zero one zero covers sequences, finance, probability and statistics, and this lesson works one worked example in each area, always with the same discipline: show the known data, name why the formula applies, write the equation in symbols, then substitute, state the unit, and interpret the result.
Here is the method, and it applies to every frame that follows. Show the known data. Say why the formula applies to it. Write the equation in symbols first, then substitute the numbers. State the unit, and interpret the result in one sentence. Two honesty rules travel with it. 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. And your centre's current brief determines the actual assessment tasks, weights, tools and deadlines; these practice sheets do not establish their official pattern or university credit.
All financial amounts, rates, fees and pay rules in the examples are invented teaching conditions. When a pay example says a fixed deduction of fifteen units, use fifteen, and do not reach for a real tax code. When a loan example says interest is charged before payment, use that order even where a familiar product does the opposite. The purpose of the stated assumptions is to make the arithmetic fully checkable, so the skill being trained is the reasoning chain, not recall of any jurisdiction's rules.
A sequence is an ordered list of numbers, and a series is their sum. An arithmetic sequence adds a fixed common difference, d, each step, like row counts going four, seven, ten. A geometric sequence multiplies by a fixed common ratio, r, each step, like a balance growing at two percent per period. The general terms are u sub n equals a plus, open, n minus one, times d, and u sub n equals a times r to the power n minus one. Keep the two models separate, because finance mixes them freely: some products compound geometrically, while others use simple interest or plain cash flows. Match the model to the specified rule.
The arithmetic sum is S sub n equals n over two, times, open bracket, two a plus, n minus one d, close bracket. The geometric sum is S sub n equals a times, open, one minus r to the n, close, over, one minus r, valid when r is not one. A geometric series with absolute r below one has a sum to infinity, a over, one minus r, which is the only case where adding forever has an answer. And the standing caution: keep a term separate from a total. The fifth term is one row; the first five rows together are a sum. Confusing them is the most common mark-loser in this topic.
The figure shows three rows containing four, seven and ten dots, so the row counts differ by a constant amount and the sequence is arithmetic with a equal to four and d equal to three. For the fifth row, the known data goes into the general term: u sub five equals four plus, open five minus one, close, times three, which is four plus twelve, so the fifth row has sixteen dots. Notice the order of the work: what is known, why the arithmetic model applies, the equation in symbols, then the substitution. The answer is a count of dots, sixteen, and it describes this row pattern, not every pattern with a three-dot difference.
Now the total. The first five rows together call for the arithmetic sum, not the fifth term. In symbols, S sub n equals n over two, times, open bracket, two a plus, n minus one d, close bracket. Substituting, S sub five equals five halves, times, open, two times four plus four times three, close, which is five halves times twenty, giving fifty dots across the five rows. Sixteen was one term; fifty is the sum. Practice sheet two point one gives a progressively harder set with worked solutions, and the same term-versus-total discipline runs through all of it.
Pause and choose. The question asks about the first six rows together, and together means a total, so the sum formula applies: S sub six equals six halves, times, open, eight plus fifteen, close, which is three times twenty-three, sixty-nine dots. The term formula would answer a different question, how many dots sit in row six alone, which is nineteen. There is no geometric ratio here, and no infinite sum, because the difference is additive and the rows do not shrink.
This unit is the arithmetic of an ordinary life, in English. Gross pay is what you earn; net pay is what arrives after deductions such as tax and insurance. Simple interest, I equals P r t, charges interest on the original amount only. Compound interest, A equals P times, one plus r, to the n, charges interest on the interest, which is exactly why it is geometric. Depreciation is a reduction in recorded value, and a fixed-percentage model multiplies by one minus r each period, though other depreciation conventions use different rules. The skill is matching the model to the stated rule, every time.
The worked pay record separates eight regular hours from two overtime hours, with their respective rates. Known: eight hours at twenty units per hour, and two overtime hours at one point five times that rate. Separate the two categories before adding. In symbols, gross pay G equals h sub r times p, plus h sub o times k times p. Substituting, eight times twenty is one hundred sixty, and two times one point five times twenty is sixty, so gross pay is two hundred twenty units. The invented fixed deduction of fifteen units comes off once: net pay N equals G minus D, two hundred twenty minus fifteen, which is two hundred five units. Practice sheet two point two gives a progressively harder set with worked solutions.
A discount of twenty percent multiplies the price by zero point eight. Two successive discounts, twenty percent and then ten percent, multiply by zero point eight times zero point nine, which is zero point seven two, a twenty-eight percent reduction, not thirty. The temptation to add the percentages is the trap this rule exists for, and the reason is structural: the second discount applies to the already-reduced price, so discounts compose multiplicatively. The same composition governs compounding in the next section, where growth factors multiply across periods for exactly the same reason.
Pause and choose. Regular pay is six times fifteen, ninety units. Overtime is four times one point five times fifteen, which is ninety units as well, so gross is one hundred eighty. The deduction of ten comes off once, leaving one hundred seventy units. The wrong options each drop one step of the method: ninety stops after regular hours, one hundred eighty forgets the deduction, and one hundred sixty treats overtime as flat time. The method is the answer, and the method keeps the categories separate to the very end.
Under a stated constant discount rate and timing assumption, the present value of a future amount is P V equals F V over, open, one plus r, close, to the n. An annuity has specified payments at regular intervals, while a loan repayment schedule may have changing amounts or an adjusted final payment, and you must specify whether interest comes before or after each payment. The label A P R is an annual rate measure whose calculation and included charges depend on the stated convention; these sheets use explicitly defined period rates instead of an unstated A P R convention. The discipline is always the same: the timing and the convention are part of the data.
A nominal twelve percent teaching rate, divided into monthly periods, gives one percent per month. But twelve monthly growths of one percent each compound: one point zero one to the twelfth power, minus one, is about twelve point six eight percent effective annual growth. Dividing the year into periods does not undo the compounding, because each period's interest earns the next period's interest. And the boundary matters: a different compounding frequency, or a fee schedule, needs a different calculation, so an effective rate is always reported with its frequency attached.
The loan-period flow takes an opening balance of one thousand, adds two percent interest, then subtracts a payment of three hundred. In symbols, B one equals B zero times, one plus r, minus M. Substituting, one thousand times one point zero two is one thousand twenty, minus three hundred leaves seven hundred twenty. The next period's interest uses seven hundred twenty, not the original principal: seven hundred twenty times one point zero two is seven hundred thirty-four point four, minus three hundred leaves four hundred thirty-four point four zero units. That one rule, interest on the running balance, is what separates a loan schedule from simple interest, and following the stated order, interest then payment, is what makes the schedule checkable. Practice sheet two point three gives a progressively harder set with worked solutions.
Pause and choose. The closing balance is four hundred thirty-four point four zero. Interest of two percent applies to the running balance of seven hundred twenty, giving fourteen point four zero of interest and a pre-payment balance of seven hundred thirty-four point four zero; the three hundred payment then leaves four hundred thirty-four point four zero. Subtracting the payment first, four hundred twenty, ignores the stated order. Using the original thousand for the interest gives four hundred twenty-four point eighty, which resurrects a balance the schedule has already left behind.
Probability measures how likely an event is, on a scale from zero to one. For equally likely outcomes, P of A is favourable over total. Mutually exclusive events cannot both happen, so their probabilities add. Independent events do not affect each other, so their probabilities multiply. The complement rule, P of not A equals one minus P of A, is often the fastest route to an answer containing at least one. Conditional probability is P of A given B equals P of A and B, over P of B, for P of B greater than zero. A union subtracts its overlap. Counting techniques supply the denominator: a permutation counts ordered selections, a combination counts unordered ones. And the standing caution: association in a supplied table does not by itself establish a cause.
The two-draw tree shows a bag with two red and one blue counter, drawn twice without replacement. To get two reds, follow the red-red path and multiply its conditional probabilities. In symbols, P of R R equals P of R one, times P of R two given R one. The first draw is red with probability two thirds. Given that, one red remains among two counters, so the second is red with probability one half. Two thirds times one half is one third. A first blue leaves only red counters, so the blue-blue path has probability zero, and the draws are not independent. Practice sheet two point four gives a progressively harder set with worked solutions.
Pause and choose. At least one red is the complement of no reds, and no reds means blue on both draws. The first draw is blue with probability one third, but then only red counters remain, so blue-blue has probability zero. Therefore at least one red has probability one minus zero, which is one: with two draws and only one blue counter, a red is certain. The complement route answered in two lines what a case-by-case sum would take longer to say, and it is the same route to use whenever you meet the words at least one.
A population is everyone you want to describe; a sample is who you actually measure. Generalising to a population needs a suitably representative sample and an honest account of selection and nonresponse, because arithmetic alone does not repair missing or systematically selected observations. A class survey can describe that class without establishing a school-wide percentage. Qualitative data are categories; quantitative data measure amounts, either discrete and counted, or continuous and measured. Bias enters through who you ask, who answers, and how the question is worded, so the data collection story is part of the data.
The histogram has intervals zero to ten and ten to thirty, with frequencies of twenty each. The widths differ, so bar height must be frequency divided by class width, which is the frequency density, D equals f over w. The first class: twenty over ten is two. The second: twenty over twenty is one. The second bar is half as high but twice as wide, and both areas represent twenty observations. Equal bar heights can conceal unequal frequencies, and equal frequencies can produce unequal heights, whenever widths differ. Keep categories in separated bars and numeric continuous intervals in a histogram. Practice sheet two point five gives a progressively harder set with worked solutions.
The mean is the total divided by the count. It uses every value, and every outlier, which is both its power and its weakness. The median is the middle value in order; moving an extreme value without changing the ordered middle may leave it unchanged. The mode 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, but the mean still measures total per observation and the median still locates the ordered middle. Neither says that most observations equal it.
Two groups contribute different totals: ten learners with mean sixty, and twenty learners with mean seventy-five. To combine them, recover each group's total first: ten sixties are six hundred, and twenty seventy-fives are one thousand five hundred. The combined mean is the total over the count: two thousand one hundred over thirty, which is seventy. The unweighted mean of sixty and seventy-five would be sixty-seven point five, which ignores the unequal counts and answers a question nobody asked, the average of groups rather than of learners. Practice sheet two point six gives a progressively harder set with worked solutions.
Pause and choose. With most incomes near forty and one at four hundred, the median best describes a typical income, because the single extreme pulls the mean far above nearly everyone. That does not make the mean wrong: it remains the right summary for total per person, which is what a payroll total divided by headcount means. The mode is not always best, and the three summaries are not interchangeable; each answers its own question. The skill being examined is choosing the summary for the question, and stating which question you answered.
The range is the largest minus the smallest, the full span, so an extreme can change it substantially. The interquartile range, Q three minus Q one, spans the central half by the stated quartile convention, and a box plot shows quartiles and a median with whisker rules that must be specified. Standard deviation summarises spread using squared deviations: for a complete population, sigma is the square root of the sum of squared deviations over n; a sample estimate divides by n minus one under that stated convention. Variance has squared units and standard deviation has the original units. A larger standard deviation means a wider spread, and two data sets can share a mean while describing completely different situations.
The box plot marks a minimum of two, a first quartile of four, a median of six, a third quartile of eight, and a maximum of twelve. The range is twelve minus two, which is ten: the full span. The interquartile range is eight minus four, which is four: the central half. Two different quantities from the same picture, answering different questions about spread. And the honesty boundary: the box does not identify every original value, the sample size, or why the spread arose. On this sheet, quartiles are medians of ordered half-lists, excluding the overall median for an odd count, with whiskers to the minimum and maximum; do not mix that convention with software-interpolated quartiles or outlier-fence whiskers. Practice sheet two point seven gives a progressively harder set with worked solutions.
The normal distribution is a continuous, symmetric, bell-shaped model, specified completely by its mean and its positive standard deviation. Those two summaries alone do not establish that observed data follow the model; they merely define it. Under a normal model, the approximate empirical rule holds: about sixty-eight percent of values lie within one standard deviation of the mean, ninety-five percent within two, and ninety-nine point seven percent within three. Those three anchors let you sanity-check any normal computation: if your answer claims a mass of data more than three standard deviations out, the arithmetic, not the data, is the suspect.
A z-score says how many standard deviations a value sits from the mean: z equals x minus mu, over sigma. Standardising is what makes two different scales comparable, because both become distances in standard deviations. For a normal model, use the cumulative area Phi of z, the probability that Z is at most z, and the symmetry relation Phi of minus z equals one minus Phi of z. An interval subtracts its two cumulative bounds. And keep the last distinction: the expected count is the sample size times the probability, and it need not be an integer, while an observed count is an integer. The model predicts smoothly; the data counts discretely.
An invented measurement model has a mean of fifty and a standard deviation of ten, and the question is the probability of a value above sixty. Standardise the threshold: z equals sixty minus fifty, over ten, which is one. The curve marks z equal to one and shades the right tail. The supplied cumulative area is Phi of one equals zero point eight four one three, so the upper tail is its complement: one minus zero point eight four one three, which is zero point one five eight seven. This is a model probability, not a guarantee that exactly fifteen point eight seven percent of a finite sample exceeds sixty; the model predicts smoothly, and the sample counts discretely. Practice sheet two point eight gives a progressively harder set with worked solutions.
Pause and choose. Standardise first: forty minus fifty over ten is minus one. The cumulative area at minus one uses symmetry: Phi of minus one equals one minus Phi of one, which is one minus zero point eight one three, zero point one five eight seven. The mirror trap gives zero point eight four one three, the area below plus one, which answers a different question. Zero point five would be the area below the mean itself, and zero point zero two two eight belongs to a threshold two standard deviations away. Standardise, then choose the tail, then subtract, in that order.
Pause for four checks. First, rows of four, seven and ten continuing with the same difference give a equal to four and d equal to three, and row five is sixteen, from u sub five equals four plus four times three. Second, twenty percent then ten percent off multiplies by zero point eight times zero point nine, which is zero point seven two, a twenty-eight percent reduction, because the second discount applies to the reduced price. Third, the combined mean is seventy because the counts weight the totals, two thousand one hundred over thirty; the unweighted sixty-seven point five averages groups instead of learners. Fourth, the threshold sixty standardises to z equal one, and the question's direction decides the tail: above sixty uses one minus Phi of one. Carry the method, known, symbols, substitution, unit, interpretation, into your next practice sheet.