Exercise 8 — Inductive Reasoning

Covers: Lesson 08 Difficulty: Intermediate

Problem 1: Evaluate the Analogy

Rate each analogy as strong or weak. Explain which criteria (from the six) support your rating.

A. “Robot-42 (Rev C, firmware v1.18, Warehouse A, polished floor) had stale sensorbar data and the root cause was SPI clock jitter. Robot-87 (Rev C, firmware v1.18, Warehouse B, concrete floor) also has stale sensorbar data. Therefore, Robot-87 probably has SPI clock jitter too.”

B. “My laptop crashed after a software update. A robot crashed after a firmware update. Therefore, the robot crash was probably caused by the firmware update, just like my laptop.”

C. “In the last 8 incidents involving estimator divergence at Hamamatsu, 7 were caused by sensorbar cable issues. This is another estimator divergence at Hamamatsu. Therefore, it’s probably a sensorbar cable issue.”

Problem 2: Mill’s Methods

Identify which of Mill’s methods is being applied, and state the conclusion.

A. | Robot | Firmware | HW Rev | Floor | Temperature | Failure? | |—|—|—|—|—|—| | R-42 | v1.18 | C | Polished | 25°C | Yes | | R-53 | v1.20 | D | Concrete | 30°C | Yes | | R-67 | v1.18 | C | Tile | 22°C | Yes | | R-71 | v1.20 | D | Polished | 28°C | Yes |

All failing robots are in Warehouse A (not shown in table but true). What method? What conclusion?

C. A robot’s motor current draw increases proportionally as ambient temperature increases from 20°C to 45°C, with a consistent linear relationship observed across 50 measurements.

D. Total system latency is 12ms. Network latency accounts for 3ms, sensor polling for 4ms, and computation for 2ms. Therefore, the remaining 3ms is attributable to actuator command transmission.

Problem 3: Probability Calculation

A. A fleet has 200 robots. 15 have hardware Rev A, 85 have Rev B, and 100 have Rev C. If a robot is randomly selected, what’s the probability it’s Rev C?

B. P(firmware bug) = 0.10, P(hardware fault) = 0.05, P(both) = 0.01. What’s P(firmware bug OR hardware fault)?

C. Two independent robots each have a 0.02 probability of failure per shift. What’s the probability that both fail in the same shift?

D. A diagnostic test for SPI failures has: sensitivity = 0.95 (P(positive | SPI failure) = 0.95), specificity = 0.90 (P(negative | no SPI failure) = 0.90). The base rate of SPI failures is 3%. If the test is positive, what’s the probability the robot actually has an SPI failure? (Use Bayes’ theorem.)

Problem 4: Identify the Causal Fallacy

Each scenario commits a causal reasoning error. Name it and explain.

A. “We deployed firmware v1.24 on Monday. On Tuesday, a completely different robot (that wasn’t updated) crashed. Therefore, the deployment process somehow destabilized the fleet.”

B. “Ice cream sales and drowning rates both increase in summer. Therefore, eating ice cream causes drowning.”

C. “Robot-42 crashed 3 times at 3 AM. 3 AM must be a problematic time for robots.”

Problem 5: Experimental Design

A fleet of robots is experiencing intermittent localization failures. You suspect either (a) a firmware timing bug, (b) floor surface changes, or (c) temperature-related sensor drift.

Design a testing plan using Mill’s methods to isolate the cause. Specify: 1. What variables you would control. 2. What you would vary. 3. How you would apply the method of difference. 4. What observations would confirm/disconfirm each hypothesis.

Problem 6: Bayes in Practice

You’re investigating a robot failure. Initially, you consider three hypotheses: - H1: SPI bus failure (prior: 20%) - H2: Software crash (prior: 50%) - H3: Power supply issue (prior: 30%)

Evidence E1: The error log shows “DMA timeout.” - P(E1 | H1) = 0.90 (SPI failure almost always causes DMA timeout) - P(E1 | H2) = 0.15 (some software crashes cause DMA timeout) - P(E1 | H3) = 0.05 (power issues rarely cause DMA timeout)

Update your beliefs using Bayes’ theorem. Which hypothesis is now most likely?

Solutions

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**Problem 1:** **A.** **Strong.** High number of relevant similarities (same hardware rev, same firmware — these are causally relevant to SPI behavior). The one dissimilarity (floor type) is probably not relevant to SPI issues. Conclusion is appropriately hedged ("probably"). **B.** **Weak.** The similarities are superficial: a laptop and a robot are very different systems. A "software update" and a "firmware update" have different characteristics. The only similarity is "device crashed after update," which is too vague to be compelling. **C.** **Strong.** Based on 8 diverse instances with 87.5% agreement. The pattern is specific (same symptom, same location, same cause in 7/8 cases). Appropriately probabilistic conclusion. **Problem 2:** **A.** Method of Agreement — All failing instances share one common factor (Warehouse A). Conclusion: Something about Warehouse A is probably causing the failures. **B.** Method of Difference — Same robot, same everything except firmware version. When firmware changes from v1.18 to v1.24, failure stops. Conclusion: The firmware (specifically the difference between v1.18 and v1.24) is probably the cause. **C.** Method of Concomitant Variation — As temperature increases, current draw increases proportionally. Conclusion: Temperature is probably a causal factor in motor current draw. **D.** Method of Residues — Known causes account for 9ms of 12ms. The remaining 3ms is attributed to the remaining factor (actuator command transmission). **Problem 3:** **A.** P(Rev C) = 100/200 = 0.50 **B.** P(firmware OR hardware) = P(F) + P(H) - P(both) = 0.10 + 0.05 - 0.01 = **0.14** **C.** P(both fail) = 0.02 × 0.02 = **0.0004** (independent events) **D.** Bayes' theorem: - P(positive | no SPI failure) = 1 - specificity = 0.10 - P(SPI failure) = 0.03 - P(no SPI failure) = 0.97 P(positive) = P(pos|SPI) × P(SPI) + P(pos|no SPI) × P(no SPI) = 0.95 × 0.03 + 0.10 × 0.97 = 0.0285 + 0.097 = 0.1255 P(SPI failure | positive) = (0.95 × 0.03) / 0.1255 = 0.0285 / 0.1255 ≈ **0.227** Only ~23%! Despite the test's 95% sensitivity, the low base rate (3%) means a positive result is still more likely to be a false positive. You need additional evidence before concluding SPI failure. **Problem 4:** **A.** False cause (non causa pro causa) — the crashed robot wasn't even updated. There's no causal mechanism connecting the deployment of one robot's firmware to another robot's crash. This is coincidence, not causation. **B.** Confounding variable — both ice cream sales and drowning increase because of a third factor (hot weather). This is the classic "correlation ≠ causation" example. **C.** Insufficient data + possible confounding — 3 crashes is too few to establish a pattern, and "3 AM" might correlate with other factors (shift change, different traffic patterns, lower staffing, temperature changes, scheduled maintenance activities). **Problem 5:** **Testing plan:** 1. **Variables to control:** Hardware revision, robot model, payload weight, route, operator procedures. 2. **Variables to vary (one at a time):** - Firmware version (v1.18 vs v1.24) — tests hypothesis (a) - Floor surface (original vs. new wax/coating) — tests hypothesis (b) - Temperature (morning 20°C vs afternoon 35°C, or use climate control) — tests hypothesis (c) 3. **Method of Difference application:** - Take one robot. Run it under identical conditions EXCEPT change one variable. - Test 1: Same robot, same conditions, firmware v1.18 → observe failure/no failure. Then firmware v1.24 → observe. - Test 2: Same robot, same firmware, original floor → observe. Then modified floor → observe. - Test 3: Same robot, same firmware, same floor, controlled temperature 20°C → observe. Then 35°C → observe. 4. **Observations:** - If failure only occurs with v1.18 and not v1.24 → firmware (a) confirmed. - If failure only occurs on new floor surface → floor (b) confirmed. - If failure correlates with temperature → sensor drift (c) confirmed. - If failure persists regardless → a different cause or combination. **Problem 6:** P(E1) = P(E1|H1)×P(H1) + P(E1|H2)×P(H2) + P(E1|H3)×P(H3) = 0.90×0.20 + 0.15×0.50 + 0.05×0.30 = 0.18 + 0.075 + 0.015 = 0.27 Updated probabilities: - P(H1|E1) = (0.90 × 0.20) / 0.27 = 0.18 / 0.27 ≈ **0.667 (66.7%)** - P(H2|E1) = (0.15 × 0.50) / 0.27 = 0.075 / 0.27 ≈ **0.278 (27.8%)** - P(H3|E1) = (0.05 × 0.30) / 0.27 = 0.015 / 0.27 ≈ **0.056 (5.6%)** **SPI bus failure (H1)** is now the most likely hypothesis at 66.7%, up from 20%. Software crash dropped from 50% to 28%. Power supply is nearly ruled out at 5.6%. The DMA timeout evidence dramatically shifted the probability toward SPI failure because DMA timeouts are highly associated with SPI issues and rarely caused by other factors.