Equity Research puzzles, solved step by step

0.648. At least two means exactly two or all three. Exactly two can happen three ways, each with chance 0.6 x 0.6 x 0.4 = 0.144, so 0.432 together. All three beat with 0.6 cubed, 0.216. Adding them gives 0.648. The complement checks it: none beat 0.064 plus exactly one 0.288 is 0.352, and 1 minus 0.352 is 0.648.

How do you avoid missing an outcome?

Think of three friends each deciding whether to come to dinner. At least two coming covers four different guest lists: each of the three possible pairs, and all three. List the outcomes by how many succeed, count the ways each count can happen, and multiply by the chance of one such way. With three stocks there are only eight results, so write them down; with more, the count of ways is a binomial coefficientThe number of ways to choose k items out of n, written n choose k; here 3 choose 2 is 3..

Is independence a fair assumption for three stocks you cover?

Usually not, and saying so is the part that sounds like an analyst. Stocks in one sector share demand, input costs and the same consensus-setting habits, so beats tend to cluster: when one beats, the others are more likely to. Clustering fattens both ends. The chance of all three beating and of none beating both rise above 0.216 and 0.064, and whether at least two beat can move either way, so the independent answer is a starting point, not a forecast.

About 31%: roughly two flags in three are false alarms. Take 10,000 companies. 200 manipulate and the screen catches 180 of them. 9,800 are clean and 4% of them, 392, are flagged anyway. Of 572 flags, 180 are real, which is 31.5%. The false alarms come from the huge clean group, so a rare problem plus a small error rate swamps the true hits.

Why is the answer not 90%?

Think of the smoke alarm in a kitchen. It goes off for every real fire, and it also goes off for burnt toast. Fires are rare and toast burns every week, so most of the times the alarm sounds, nothing is on fire. The 90% tells you how the screen treats a manipulator; the question asks how often a flag is right, and that depends on how many clean companies are standing there to be wrongly flagged. Those are two different conditional probabilities, and swapping them is the whole trap.

How do you set it up so the arithmetic is easy out loud?

Pick a round population and count people, not probabilities. With 10,000 companies the 2% who manipulate are 200, and 90% of them, 180, get flagged. The clean 9,800 produce 4% false flags, 392. Once the tree is drawn, the answer is just flagged guilty over all flagged: 180 over 572. This counting method is called natural frequenciesStating a probability problem as counts out of a round population, such as 180 out of 10,000, instead of as percentages. It makes conditional reasoning much easier to follow., and it is far harder to get wrong than juggling Bayes' formula in your head.

What does this mean for how an analyst uses a screen?

A flag is evidence, not a verdict. It moves the odds from 2 in 98 to 180 in 392, a jump of 22.5 times, which is exactly 90% divided by 4%. That is why a flag should start a deeper read of the notes to the accounts, never end one. If a second, independent check with the same accuracy also flags the company, the odds multiply again and the chance rises to about 91%. The limitation is the word independent: two ratio screens built on the same receivables line will fail together.