Probability Paradoxes

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?

IV 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: — (theory: 33%)

When you SWITCH

0 / 0 wins

Win rate: — (theory: 67%)

Games played: 0

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:

I Boy or Girl?

II The Birthday Problem

III Simpson's Paradox

IV The Monty Hall Problem

V Six Degrees of Separation

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%