Puzzle #1: Powerball
Back in the day, before the Powerball and Pick 6, people used to bet on dice. One game involved rolling 3 dice and betting on the sum. Everyone assumed that the probability of the sum equaling 9 was equal to the probability of the sum equaling 10 because there are 6 ways to get each sum: 10 = (6 + 3 + 1) = (6 + 2 + 2) = (5 + 4 + 1) = (5 + 3 + 2) = (4 + 4 + 2) = (4 + 3 + 3) and 9 = (6 + 2 + 1) = (5 + 3 + 1) = (5 + 2 + 2) = (4 + 4 + 1) = (4 + 3 + 2) = (3 + 3 + 3).
One gambler used to clean up because he figured out that the sum of 10 was actually more likely than a sum of 9. What did this gambler realize?
Puzzle #2: Sophomore Slump
The phrase “sophomore slump” is used to describe a number of phenomena. One phenomenon occurs in music when bands have a hugely successful first release followed by a mediocre second release. A similar phenomenon occurs in sports when the player voted “rookie of the year” has a mediocre second season. Can you use probability (not psychology) to explain the sophomore slump? (We have not touched on the reason, but it is probably something you discussed in Stats or Methods.)
Puzzles 3 – 5 show how a hidden third variable can mess with statistics. The puzzles are challenging, so I’ve broken them down into steps. (The puzzles all have a similar structure, so if you can understand one of the puzzles, you’ll likely understand them all.)
Puzzle #3: Hospitals
Consider the recovery rates at two hospitals:
Survived Died Total Recovery Rate
Hospital A 800 200 1000 80%
Hospital B 900 100 1000 90%
From these stats, Hospital B would appear to be the more successful hospital.
But consider the recovery rates for the intensive care units and the general care units at each hospital.
ICU statistics:
Survived Died Total Recovery Rate
Hospital A 210 190 400
Hospital B 30 70 100
General care statistics:
Survived Died Total Recovery Rate
Hospital A 590 10 600
Hospital B 870 30 900
Does Hospital B still appear more successful? If not, what’s going on?
Step 1: Calculate the ICU Recovery Rates.
Step 2: Calculate the General Care Recovery Rates.
Step 3: Explain what’s going on (identify the third variable). Remember a third variable is something that varies with the independent variable, in this case, it’s a key difference between the patients in the two hospitals.
Puzzle #4: Professional School Admissions
Let’s try that again, but this time looking at law school and medical school admission rates for men and women:
Overall Rates (Law + Medical):
Applied Accepted Acceptance Rate
Men 430 226
Women 390 177
Law School
Applied Accepted Acceptance Rate
Men 250 190
Women 150 117
Medical School
Applied Accepted Acceptance Rate
Men
Women
Step 1: Calculate the overall acceptance rates for men and women.
Step 2: Calculate the law school acceptance rates for men and women.
Step 3: Determine the numbers of men and women who applied and were accepted to medical school.
Step 4: Calculate the medical school acceptance rates for men and women.
Step 5: Looking at the overall acceptance rates, which gender does better? Looking at the rates for each school, which gender does better?
Step 6: Identify the third variable that is causing this switch.
Puzzle #5: Batting Averages
2017 Hits At Bat Batting Avg.
Player A 12 48
Player B 104 411
2018 Hits At Bat Batting Avg.
Player A 183 582
Player B 45 140
Combine the two years:
Overall (2017 + 2018) Hits At Bat Batting Avg.
Player A
Player B
Step 1: Calculate the total number of Hits and At Bats for each players over the two years.
Step 2: Calculate the 2017 Batting Averages.
Step 3: Calculate the 2018 Batting Averages.
Step 4: Calculate the 2017 + 2018 Batting Averages.
Step 5: Who is the better player when you look at the combined averages? Who is the better player when you look by each year?
Step 6: Identify the third variable that is causing this switch.