Paradoxes

Abdullah Al Mahmud

Simpson’s Paradox

Who is better: Sayem or Siyam?

Exam Sayem Siyam
Exam-1 \(\frac{63}{90} = 70 \%\) \(\frac{8}{10}=80 \%\)
Exam-2 \(\frac{4}{10} = 40 \%\) \(\frac{45}{90} = 50 \%\)

Look again

Who is really better?

Exam Sayem Siyam
Exam-1 \(\frac{63}{90} = 70 \%\) \(\frac{8}{10}=80 \%\)
Exam-2 \(\frac{4}{10} = 40 \%\) \(\frac{45}{90} = 50 \%\)
Combined \(\frac{67}{100} = 67 \%\) \(\frac{53}{100} = 53 \%\)

Batting average

Year Jeter Justice
1995 \(\frac{12}{48} = 0.25\) \(\frac{104}{411}=0.253\)
1996 \(\frac{183}{582} = 0.314\) \(\frac{45}{140} = 0.270\)

Take a second look

Year Jeter Justice
1995 \(\frac{12}{48} = 0.25\) \(\frac{104}{411}=0.253\)
1996 \(\frac{183}{582} = 0.314\) \(\frac{45}{140} = 0.321\)
Combined \(\frac{195}{630} = 0.310\) \(\frac{149}{551} = 0.270\)

Compare angularly, most matches simultaneously

But NOT SO Straightforward

Stone removal from kidney

Stone Treatment A
(Open Surgery)		 Treatment B
(Puncture)
Small Group 1

\(\frac{81}{87} = 93 \%\)		 Group 2

\(\frac{234}{270}=87\%\)
Large Group 3

\(\frac{192}{263} = 73\%\)		 Group 4

\(\frac{55}{80} = 69 \%\)

But NOT SO Straightforward

Stone Treatment A
(Open Surgery)		 Treatment B
(Puncture)
Small Group 1

\(\frac{81}{87} = 93 \%\)		 Group 2

\(\frac{234}{270}=87\%\)
Large Group 3

\(\frac{192}{263} = 73\%\)		 Group 4

\(\frac{55}{80} = 69 \%\)
Both \(\frac{273}{350} = 78\%\) \(\frac{289}{350} = 83 \%\)
  • Statistics never lies!

Problem of Points

  • Suppose that two players, A and B, contribute equally to a stake of $60.
  • They agree that the first player who makes 3 points shall win the entire stake.
  • After A has won 2 points, and B has won 1, they agree to stop.
  • How should the stake of 60 dollars be divided?
  • A = 40 & B = 20?
  • If another match is played
  • If A wins A = 60, B = 0
  • If B wins A = 30, B = 30
  • On average, A = 45, B = 15

All Windows to South

Can you make a square house with windows on all four sides, each window having a view to the south?

Answer

See on map

Is USA to the North of Russia

Then to which side of Russia is USA?

Back Home

Can a man set out from his house, walk five miles due south, five miles due west, and five miles due north-and find himself back home?

Salary Raises

  • Salary is $1000/year, to be paid half-yearly.
  • Which do you prefer: a raise of $150 per year or a raise of $50 every half-year?
Answer

Bottle-Cork

A bottle and its cork cost together $1.10. The bottle costs a dollar more than the cork. How much does the bottle cost?

  • $1?

Confusing Questions

Meeting Trains

An express leaves New York for Boston at the same time that a local leaves Boston for New York. The express travels at the rate of 50 miles per hour, the local at the rate of 30 miles per hour.

Which is farther from New York when they meet?

  • Same!

Red-green

Which is RED and which is GREEN?

RED

GREEN

Northia & Southia

  • Northia and Southia-had an agreement whereby a Northian dollar was worth a dollar in Southia, and vice versa.
  • But one day the government of Northia decreed that thereafter a South ian dollar was to be worth but ninety cents in Northia. The next day the South ian government, not to be outdone, decreed that thereafter a Northian dollar was to be worth but ninety cents in Southia.
  • Now a bright young man lived in a town which straddled the border between the two countries. He went into a store on the Northian side, bought a ten-cent razor, and paid for it with a Northian dollar. He was given a Southian dollar, worth ninety cents there, in change.
  • He then crossed the street, went into a Southian store, bought a ten-cent package of blades, and paid for it with the South ian dollar. There he was given a Northian dollar in change. When the young man returned home, he had his original dollar and his purchases. And each of the tradesmen had ten cents in his cash-drawer.
  • Who, then, paid for the razor and blades?

Statistical Paradoxe

Speed Paradox

A man drove his car 1 mile to the top of a mountain at the rate of 15 miles per hour. How fast must he drive 1 mile down the other side in order to average 30 miles per hour for the whole trip of 2 miles?

  • Use HM when denominator of rate is constant
  • Infinite speed (0 second)

Probability Paradoxes

Zero vs Infinitesimal Probability

Illusions

Checker Shadow Illusion

Problem

Solution

Impossible Objects

Impossible Cube

See More

Real World Example

Real world is not what we think it to be

See more here

References