Which is better – wealth or ability? Fred Mosteller posed this question in his classic 1965 small compendium Fifty Challenging Problems in Probability, in the context of the Gambler’s Ruin puzzle. Two players, M and N, engage in a game in which $1 is transferred from one player to the other at each play. PlayerContinue reading “Puzzle – Gambler’s Ruin”
Several decades ago, the dominant therapies for lung cancer were radiation, which offered better short-term survival rates, and surgery, which offered better long-term rates. A thought experiment was conducted in which surgeons were randomly assigned to one of two groups and asked whether they would choose surgery.
Group 1 was told: The one-month survival rate is 90%.
Group 2 was told: There is 10% mortality in the first month.
Yes, the two statements say the same thing. What did the two physician groups choose?
Our problem of the week is an ethical dilemma, posed by the New England Journal of Medicine to its readers 10 days ago. Volunteers contributed DNA samples to investigators building a genetic database for study, on condition the data would be deidentified and kept confidential and that they themselves would not learn results. Should participantsContinue reading “Problem of the Week: Notify or Don’t Notify?”
QUESTION: A gambler playing against the “house” in a game like roulette or slots adopts the rule “Play until you win a certain amount, then stop.” Will this ensure against player losses? What will be its effect on the house’s profit? ANSWER: Some look at this rule and figure that it rules out player losses andContinue reading “Problem of the Week: A betting puzzle”
QUESTION: A friend tosses two coins, and you ask “Is one of them a heads?” The friend replies “Yes.” What is the probability that the other is a heads? ANSWER: One-third. There are four ways the coins could have landed originally: HH: 0.25 probability HT 0.25 probability TH 0.25 probability TT Continue reading “Problem of the Week: The Second Heads”
Question: A baseball team is comparing two of its hitters, Hernandez and Dimock. Hernandez hit .250 in 2017 and .275 in 2018. Dimock did worse in both years – .245 in 2017 and .270 in 2018. Overall, though, Dimock hit better across the two years, .263 versus .258 for Hernandez. How can this be? Answer:Continue reading “Problem of the Week: Simpson’s Paradox – baseball”
Question: You work for an internet real-estate company, building statistical models to predict home price on the basis of square footage, number of bedrooms, number of bathrooms, property type (single family home, townhouse, multiplex), and age. Surprisingly, you find the coefficient for bedrooms is negative, meaning that adding bedrooms decreases value. What might account forContinue reading “Problem of the Week: The Value of Bedrooms”
Question: You have a supervised learning task with 30 predictors, in which 5% of the observations are missing. The missing data are randomly distributed across variables and records. If your strategy for coping with missing data is to drop records with missing data, what proportion of the records will be dropped? Is the assumption ofContinue reading “Problem of the Week: Missing Data”
Your country is at war, and an enemy plane has crashed on your territory. It bears the number 60, and a spy has told you that the aircraft are numbered serially. Can you make a guess about the total number of aircraft the enemy has produced? Solution: This problem is one of those published byContinue reading “Problem of the Week: Probability”