Probability, Statistics, and Engineering Economy

The last third of the MSTE math coverage rewards speed more than insight. Counting, probability, statistics, and engineering economy are formula-driven: once you match the situation to the right formula, the number falls out. This lesson gives you those formulas and walks each one to an exact answer so you can bank these items in seconds and save your thinking time for the harder surveying problems.

Counting: permutations and combinations

Use a permutation when order matters (arranging, ranking, filling distinct posts) and a combination when order does not matter (choosing a committee or a sample). The formulas are P(n, r) = n! / (n - r)! and C(n, r) = n! / [r!(n - r)!].

Worked example (permutation): Three distinct posts (foreman, deputy, recorder) are filled from 9 volunteers. Since each post is different, order matters: P(9, 3) = 9 x 8 x 7 = 504 ways. Worked example (combination): Choosing 3 test cylinders from 8 does not rank them, so C(8, 3) = (8 x 7 x 6) / (3 x 2 x 1) = 336 / 6 = 56 ways. The trap is picking the permutation count 336 when order is irrelevant.

Probability rules

Probability runs from 0 (impossible) to 1 (certain). Four rules cover almost every board item:

• Complement: P(not A) = 1 - P(A).
• Addition: P(A or B) = P(A) + P(B) - P(A and B). If A and B are mutually exclusive, the last term is 0.
• Multiplication: for independent events, P(A and B) = P(A) x P(B).
• Conditional: P(A given B) = P(A and B) / P(B). Drawing without replacement makes the second draw conditional on the first.

Worked example (complement): If a slab passes its strength test with probability 0.92, it fails with probability 1 - 0.92 = 0.08. Worked example (without replacement): A bin holds 6 good and 4 defective bolts. Draw two without replacing: P(both good) = (6/10)(5/9) = 30/90 = 1/3. Assuming replacement instead gives (6/10)(6/10) = 9/25, a classic slip.

Descriptive statistics

Four center-and-spread measures show up constantly. The mean is the sum divided by the count. The median is the middle value after sorting (average the two middle values when the count is even). The mode is the most frequent value. The range is maximum minus minimum. The standard deviation measures spread: for a population, sigma = sqrt[ sum(x - mean)^2 / N ].

Worked example (standard deviation): For 14, 18, 20, 22, 26 MPa, the mean is 100/5 = 20. The squared deviations are 36, 4, 0, 4, 36, summing to 80. Variance = 80/5 = 16, so sigma = sqrt(16) = 4 MPa. Forgetting the square root leaves you at the variance 16, the most common error. Always sort before reading a median: the unsorted middle value is a distractor, not the answer.

The normal distribution concept

Many measured quantities (concrete strength, survey errors) follow a bell-shaped normal distribution, symmetric about its mean, where the mean, median, and mode coincide. You only need the empirical (68-95-99.7) rule: about 68% of values lie within one standard deviation of the mean, about 95% within two, and about 99.7% within three. If cube strengths average 28 MPa with sigma 2 MPa, roughly 95% of cubes fall between 24 and 32 MPa.

Engineering economy formulas

Money has a time value, so a peso today is not a peso next year. Match the cash-flow pattern to its factor:

Concept	Formula
Simple interest	I = P i n; total F = P(1 + i n)
Compound interest	F = P(1 + i)^n
Present worth (single)	P = F / (1 + i)^n
Annuity present worth	P = A[1 - (1 + i)^(-n)] / i
Annuity future worth	F = A[(1 + i)^n - 1] / i
Capital recovery	A = P i / [1 - (1 + i)^(-n)]
Sinking fund	A = F i / [(1 + i)^n - 1]
Rate of return	ROR = annual profit / invested capital
Break-even	units = fixed cost / (price per unit - variable cost per unit)

The capital-recovery factor turns a present amount into equal annual payments (loan amortization); the sinking-fund factor turns a future target into equal annual deposits. They are the same idea run in opposite directions.

Worked example (simple versus compound): Borrow P120,000 at 9% for 4 years. Simple interest gives I = 120,000(0.09)(4) = P43,200, so you repay P163,200. Compounding instead, F = 120,000(1.09)^4 = P169,390, which is why lenders prefer it. Worked example (compound future worth): P35,000 at 12% compounded annually for 2 years grows to 35,000(1.12)^2 = 35,000(1.2544) = P43,904; the simple-interest figure of P43,400 is the tempting wrong answer. Worked example (present worth): To have P66,150 in 2 years at 5%, deposit P = 66,150 / (1.05)^2 = 66,150 / 1.1025 = P60,000 today. Worked example (annuity): Repaying a loan with 5 year-end payments of P20,000 at 10%, the amount borrowed today is P = 20,000[1 - (1.10)^(-5)] / 0.10 = 20,000(3.79079) = P75,815.74. Confusing this with the future worth of the same deposits, 20,000(6.1051) = P122,102, is the standard trap.

Depreciation

Three methods appear on the board. Straight line (SL) spreads the depreciable base evenly: annual charge = (first cost - salvage) / life. Sum-of-the-years-digits (SYD) front-loads it: year-k charge = [(remaining life at start of year) / SYD] x (first cost - salvage), where SYD = 1 + 2 + ... + n. Declining balance (DB) applies a fixed rate to the shrinking book value: book value after k years = first cost x (1 - rate)^k.

Worked example (SL versus SYD): Equipment costs P600,000, salvage P60,000, life 5 years, so the depreciable base is P540,000. SL gives 540,000/5 = P108,000 every year. SYD has sum 1+2+3+4+5 = 15, so the first-year charge is (5/15)(540,000) = P180,000, much larger. Swapping the two methods is the classic error. Worked example (DB): A P250,000 machine at a 20% declining-balance rate has a book value after 2 years of 250,000(0.80)^2 = 250,000(0.64) = P160,000.

Exam-day strategy

• First decide order versus no order: order matters means permutation, order does not means combination. Writing "does order matter?" on your scratch paper prevents the P-for-C swap.
• Sort every data set before you read off a median, and never confuse the variance with the standard deviation: the standard deviation is the square root.
• For interest problems, read whether it says "simple" or "compounded"; the two answers are always both in the choices to catch you.
• Match the annuity to its direction: paying off a present amount uses the capital-recovery form, saving toward a future amount uses the sinking-fund form.
• On depreciation, SYD front-loads (biggest charge in year one) while SL is level; if the item asks for the first year and the answers differ, SYD gives the larger number.