Probability, Statistics, and Engineering Economy

The last slice of the 25% Mathematics group is the most practical: counting, probability, statistics, and engineering economy. These items reward a clean formula and careful bookkeeping rather than heavy theory, so they are among the fastest points on the paper. Engineering economy in particular carries into the Allied Subjects group, where the same present-worth and depreciation ideas return. Learn the handful of formulas below cold and you turn a whole block of the exam into reliable marks.

Counting: permutations and combinations

The first decision is always the same: does order matter? If it does, use a permutation; if it does not, use a combination.

Tool	Formula	Use when
Permutation	P(n, r) = n!/(n - r)!	Arrangements, rankings, seat orders
Combination	C(n, r) = n!/[r!(n - r)!]	Committees, samples, unordered picks

Worked example (permutation): The number of ways to award first, second, and third place among 8 finalists is P(8, 3) = 8(7)(6) = 336. Worked example (combination): The number of ways to form a 3-member committee from 9 volunteers is C(9, 3) = 9(8)(7)/(3 times 2 times 1) = 504/6 = 84. Notice that P(9, 3) = 504 is exactly the combination times 3!, which is why treating an unordered choice as ordered overcounts.

Probability rules

Every probability is favorable outcomes over total outcomes, a number between 0 and 1. Four rules cover most items.

Rule	Statement
Complement	P(not A) = 1 - P(A)
Addition	P(A or B) = P(A) + P(B) - P(A and B)
Multiplication (independent)	P(A and B) = P(A) times P(B)
Conditional	P(B given A) = P(A and B)/P(A)

Worked example (addition): Drawing one card from a 52-card deck, P(heart or king) = 13/52 + 4/52 - 1/52 = 16/52 = 4/13. You subtract the king of hearts once so it is not double counted. Worked example (multiplication): Drawing two marbles with replacement from 4 white and 6 black, P(both white) = (4/10)(4/10) = 4/25. Without replacement the second draw would be 3/9, giving 2/15 instead, so read the "with or without replacement" wording carefully. Worked example (conditional): If 60% of a class passed Math and 45% passed both Math and Physics, then P(Physics given Math) = 0.45/0.60 = 0.75.

Descriptive statistics

Three measures locate the center of a data set, and one measures its spread.

• The mean is the sum divided by the count.
• The median is the middle value once the data are sorted; for an even count it is the average of the two middle values.
• The mode is the value that appears most often.
• The standard deviation measures spread. For a full population it is sqrt of the average squared deviation from the mean.

Worked example (mean): For 6, 9, 11, 14, 20 the mean is 60/5 = 12. Worked example (median): For 5, 8, 11, 14, 17, 22 the median is (11 + 14)/2 = 12.5. Worked example (standard deviation): For 2, 4, 6, 8, 10 the mean is 6, the squared deviations are 16, 4, 0, 4, 16 summing to 40, the variance is 40/5 = 8, and the population standard deviation is sqrt(8) = 2 sqrt(2), about 2.83.

The normal distribution

The normal distribution is the bell-shaped curve that many natural measurements follow. It is symmetric about its mean, and for a normal curve the mean, median, and mode coincide. The empirical rule gives the spread you should memorize: about 68% of the data lies within one standard deviation of the mean, about 95% within two, and about 99.7% within three.

Engineering economy: interest and worth

Money has a time value, so a peso today is worth more than a peso next year. The rates below are illustrative only.

Quantity	Formula
Simple interest	I = P i n; future value F = P(1 + i n)
Compound future value	F = P(1 + i)^n
Present worth	P = F/(1 + i)^n
Present worth of an annuity	P = A[1 - (1 + i)^-n]/i

Worked example (simple interest): The simple interest on PHP 35,000 at 8% for 3 years is I = 35,000(0.08)(3) = PHP 8,400. Worked example (compound): PHP 50,000 at 6% compounded annually for 4 years grows to F = 50,000(1.06)^4 = PHP 63,123.85, well above the PHP 62,000 you would get from simple interest, which is the classic trap. Worked example (annuity): The present worth of PHP 6,000 paid at the end of each year for 5 years at 8% is P = 6,000[1 - (1.08)^-5]/0.08 = PHP 23,956.26.

Engineering economy: depreciation, return, and break-even

An asset costing C with salvage value S over life n loses value each year, and three methods appear on the board.

Method	First or nth-year charge
Straight line	(C - S)/n each year
Sum-of-years-digits (SYD)	(remaining life/sum of digits)(C - S)
Declining balance	book value times a fixed rate each year

Worked example (straight line): A PHP 420,000 machine with a PHP 60,000 salvage over 6 years depreciates by (420,000 - 60,000)/6 = PHP 60,000 per year. Worked example (SYD): For PHP 240,000 with PHP 30,000 salvage over 5 years, the digits sum to 5 + 4 + 3 + 2 + 1 = 15, so the first-year charge is (5/15)(210,000) = PHP 70,000. Worked example (declining balance): A PHP 600,000 vehicle losing 25% of book value yearly is worth 600,000(0.75)^2 = PHP 337,500 after 2 years.

Two more measures close out the topic. The rate of return is annual profit divided by investment: a PHP 45,000 net profit on a PHP 250,000 investment is 45,000/250,000 = 18%. The break-even quantity is fixed cost divided by the contribution margin (price minus variable cost per unit): with PHP 40,000 fixed cost and a PHP 100 margin, break-even is 40,000/100 = 400 units.

Exam-day strategy

• Settle the order question first: if rearranging the same items counts as a new outcome, it is a permutation; if not, it is a combination. This one check kills the most common counting error.
• In the addition rule, always subtract the overlap P(A and B). Forgetting it inflates the probability and is the classic mistake on "or" items.
• Read "with replacement" versus "without replacement" before you multiply, because the second factor changes.
• Keep simple and compound interest apart: simple interest grows the principal linearly, compound interest raises (1 + i) to the nth power, and the compound result is always larger for n greater than 1.
• For SYD, sum the years first (n(n + 1)/2 is a fast shortcut) and apply the depreciable base C - S, not the full cost.
• For declining balance, multiply the book value by the retention factor each year; do not subtract a fixed percentage of the original cost, which is a different and wrong calculation.