# Mathematical Structure of Probability Space — 1

# 0. Motivation

Honestly, I find the ubiquitousness of differences in probabilities (“… led to a 3.5% point increase …”) — in news media, Social Sciences and Data Science — an indication of thoughtlessness. I’ve had people tell me to add probabilities for event sets that do not belong to a partition.

# 1. Why do we need to compare probabilities?

Is the “algebra” of probabilities worth understanding?

# Here are some questions that arise frequently:

- When is a dollar not a dollar? When is the same quantitative change not of the same “value”? E.g., if I, with say a 6 figure wealth, give a dollar to an indigent in Redwood City with possibly single figure wealth, do we both perceive the same change? Why not? How is this related to income or wealth distribution?
- Similarly, how do I compare the effect of changing the college admission rate for White children from 0.2 to 0.25 to that for Black children from 0.05 to 0.1? In both cases the “percentage point” change is 5%. Are they “just as impactful”? Why or why not? How about comparing the effect of changing the college admission rate for White children from 0.8 to 0.9 to that for Black children from 0.5 to 0.6?
- How do we evaluate “For a given infraction, the percentage of students punished that are Black is 37%”, given that the Black %age of the student population is 11%?
- The 6-month “symptom-free” rate for a disease increased from p(O|C) = 60%, to p(O|T) = 75% under a specific snake-oil treatment. (Where O = Outcome, C = Control and T = Treatment.) How would you characterize the effect of treatment? (Or Jessamy’s version: The rate of disciplinary actions for perceived student infractions decreased from 75%, to 60% under a specific “wise” classroom intervention. How would you characterize the effect of treatment?)
- The complement of the above: The 6-month recidivism rate for a disease decreased from p(O’|C) = 40%, to p(O’|T) = 25% under the snake-oil treatment. How would you characterize the effect of treatment? Is your answer consistent with the above?
- “Diff in diff”: Under a “wise” classroom intervention, the success rate for Black students increased from 0.6 without intervention to 0.75. The success rate for White students increased from 0.7 to 0.84. Was this treatment inequity reducing?
- “Diff and diff”: Research based evidence of scientifically rigorous studies show that 1 year of a school mentoring program increased White students’ sense of belonging from 85% (Control) to 93.5% (Treatment). Given only this data, what can you reasonably predict (state simplifying assumptions if needed) White students’ sense of belonging at the end of a second year of the same school mentoring program? What hypotheses do you have if the answer is not what you predict?
- If the probability that a self-identified Male will click on an ad for a handbag is 0.3 and the probability that a 45–55 year old will click on an ad for a handbag is 0.4, what is the probability that a 45–55 year old Male will click on an ad for a handbag?
- Is the unit interval (probability space) with the “obvious” Euclidean metric geodesically complete? Won’t you fall off the ends if you keep walking?

I hope that the above simple examples sufficiently motivate the need to understand probabilities and have provided some food for thought.

See later for more complex examples.