Probability Paradoxes

The Monty Hall Problem

Three doors, one car, two goats. The host opens a goat door. Should you switch?
Tre porte, una macchina, due capre. Il conduttore apre una porta. Conviene cambiare?

Pick a door!

?Door 1

?Door 2

?Door 3

Your results: (play at least 10 rounds!)

When you STAY

0 / 0 wins

Win rate: —

When you SWITCH

0 / 0 wins

Win rate: —

Games played: 0

I Boy or Girl?

A parent says: "I have two children. At least one is a boy." What is the probability both are boys?
Un genitore dice: "Ho due figli. Almeno uno è maschio." Qual è la probabilità che entrambi siano maschi?

What do you think? Pick your answer:

II The Birthday Problem

How many people needed in a room for a >50% chance of a shared birthday?
Quante persone servono per avere più del 50% di probabilità di un compleanno condiviso?

Your guess: How many people are needed?

III Simpson's Paradox

A trend in every subgroup can reverse when data is combined.
Una tendenza in ogni sottogruppo può invertirsi quando i dati vengono aggregati.

Berkeley 1973: Look at the overall admissions data.

	Men	Women
Applied	8,442	4,321
Admitted	44%	35%

Based on this data, do you think there was gender discrimination?

Now look at the data department by department:

Dept	Men applied	Men admit	Women applied	Women admit
A	825	62%	108	82%
B	560	63%	25	68%
C	325	37%	593	34%
D	417	33%	375	35%
E	191	28%	393	24%
F	272	6%	341	7%

In 4 out of 6 departments, women have higher admission rates! Women applied more to competitive departments with low rates for everyone.

Now do you still think there was discrimination?

V Six Degrees of Separation

Any two people on Earth are connected through at most ~6 intermediate acquaintances.
Qualsiasi persona al mondo può essere raggiunta attraverso circa 6 conoscenti intermedi.

How many steps do you think it takes to connect any two people on Earth through acquaintances?

The power of exponential growth: If each person knows people:

Steps	People reachable	vs World (8B)

Network simulation: Find the shortest path between two random people.

Nodes: Contacts:

The Monty Hall Problem

I Boy or Girl?

II The Birthday Problem

III Simpson's Paradox

V Six Degrees of Separation

Programming Tutorial

What you'll learn:

Teacher Dashboard

Students

I Boy or Girl — Initial Guesses

II Birthday Problem

III Simpson's Paradox — Votes

IV Monty Hall — Class Results

V Six Degrees — Guesses

Teacher Dashboard

Dept	Men applied	Men admit	Women applied	Women admit
A	825	62%	108	82%
B	560	63%	25	68%
C	325	37%	593	34%
D	417	33%	375	35%
E	191	28%	393	24%
F	272	6%	341	7%

Dept	Men applied	Men admit	Women applied	Women admit
A	825	62%	108	82%
B	560	63%	25	68%
C	325	37%	593	34%
D	417	33%	375	35%
E	191	28%	393	24%
F	272	6%	341	7%

Dept	Men applied	Men admit	Women applied	Women admit
A	825	62%	108	82%
B	560	63%	25	68%
C	325	37%	593	34%
D	417	33%	375	35%
E	191	28%	393	24%
F	272	6%	341	7%