The Little Blue Reasoning Book - Brandon Royal [20]
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PROBABILITY TREES
Tip #14: The end branches of a probability tree must total to 1, which is equal to the aggregate of all individual probabilities.
Exhibit 3.6 illustrates the probabilities associated with each event. Note that probabilities always total to 1, if we add the probabilities at the endpoints (i.e., 8 × 1⁄8 = 1.0). Each endpoint equals 1⁄8, which is the resultant probability of three consecutive tosses of a coin (i.e., 1⁄2 × 1⁄2 × 1⁄2 = 1⁄8 ).
WEIGHTED RANKING
Tip #15: Weighted ranking is a tool for finding solutions using a weighted average. To calculate weighted average, we multiply each event by its associated weight and total the results. In the case of probabilities, we multiply each event by its respective probability and total the results.
Snapshot
The weighted average concept is actually quite intuitive. To find a weighted average, we multiply events by their respective weight and total the results. Events are the things that we wish to rate, rank, or judge. Weights refer to the amount of emphasis we want to attribute to each event and are commonly expressed as percentages, fractions, decimals, or probabilities. The beauty of weighted average is that we can assign different weights based on the relative importance of events — the more important the event, the more weight it is given.
Below is the weighted average formula for two events:
Weighted Average = (Event1 × Weight1) + (Event2 × Weight2)
An alternative format is:
Exam Time
A student scores 60 out of 100 points on his midterm exam and 90 out of 100 points on his final exam. If the exams are both weighted equally, counting for 50% of the student’s final course grade, then what is his course grade?
Based on the same information above, what is the student’s final course grade if the midterm exam is weighted 40% and the final exam is weighted 60%?
Note that the weights above could also be expressed using fractions or decimals:
Hiring and promotion decisions are classic examples of situations in which subjective influences can override an objective decision-making process. Weighted ranking therefore presents a method to quantify decision opportunities.
Consider a company with ten salespersons, one of whom is to be named National Sales Manager. As depicted in Exhibit 3.7, the ten candidates are first ranked from 1 to 10 (10 being the highest rating) across three criteria.
The three criteria — technical skills, people skills, and track record — are weighted using the weights of 0.2, 0.3, and 0.5, respectively (see Exhibit 3.8). Note that instead of using decimals (0.2, 0.3, 0.5), we could also use percents (i.e., 20%, 30%, 50%), fractions (i.e., 2⁄10, 3⁄10, 5⁄10 ), or even whole numbers such as 2, 3, and 5. Based on the results from weighting the data (per Exhibit 3.9), Patricia receives the highest ranking while George gets the next highest ranking.
The weights used will typically add up to 1 or 100%, as is the case when dealing with percentages, fractions, decimals, or probabilities. Sometimes problems will use arbitrary weights which are not equal to 1.
Exhibit 3.7 – Performance of Salespersons
Exhibit 3.8 – Performance Using Weighted Average
Exhibit 3.9 – Ranking of Salespersons
Chess
In chess, a pawn is worth one point, a knight or bishop is worth three points, a rook is worth five points, and a queen is worth nine points. Player A has two rooks, a knight, and three pawns. Player B has a bishop, four pawns, and a queen. Who is ahead and by how much?
Answer: Both players are tied at 16 points each.
Sweet Sixteen
On her sixteenth birthday, Jane received $500 from each of her two uncles. Both amounts had been deposited in two local banks, one bank paying 6% per annum and the other paying 7% per annum. How much in total did she earn from these two investments over the course of exactly one year?
Problem 11: Investor
An investor is looking at three different investment possibilities. The first investment opportunity has a