The Mathematics of Game Shows - Patterns in Practice

The expecte

This frequency chart s
ways to get each resu

# prizes # ways
4
2 1
1 6
0 8
9
TOTAL 24

ed value is…

shows the number of
ult.

The total number of
prizes is
4x1+2x6+1x8
… 24 prizes and 24
ways. Divide to
find…

Hey, it’s 1!

A Second

Reconsider the proble
it. (Sue Sylvester, from

ABCD BACD
ABDC BADC
ACBD BCAD
ACDB BCDA
ADBC BDAC
ADCB BDCA

d Opinion

em from how Sue sees
m Glee.)

CABD DABC
CADB DACB
CBAD DBAC
CBDA DBCA
CDAB DCAB
CDBA DCBA

A Second

Light up all the places
game) is correctly plac

ABCD BACD
ABDC BADC
ACBD BCAD
ACDB BCDA
ADBC BDAC
ADCB BDCA

d Opinion

where B (the Glee
ced.

CABD DABC
CADB DACB
CBAD DBAC
CBDA DBCA
CDAB DCAB
CDBA DCBA

A Second

Glee is won one-fourth
time… and all the prize

ABCD BACD
ABDC BADC
ACBD BCAD
ACDB BCDA
ADBC BDAC
ADCB BDCA

d Opinion

h (6 out of 24) of the
es are like that.

CABD DABC
CADB DACB
CBAD DBAC
CBDA DBCA
CDAB DCAB
CDBA DCBA

It’s alw

With four prizes, each i
4 x (1

With five prizes, each is
With n prizes, each is w

n x (1
Clever methods can

ways 1!

is won 1/4 of the time.
1/4) = 1
s won 1/5 of the time.
won 1/n of the time.
1/n) = 1
n “beat” enumeration.

Two Ext

1.  As the number of p
happens to the p
nothing at all?

2.  The mean (average
away is always 1
the standard dev

These two problems h
which this slide is too s

tensions

prizes grows, what
probability of winning

e) number of prizes given
1. What happens to
viation?

have great conclusions,
small to contain…

(We’ll be right back.)

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The #1 Game

PLIN

e on TPIR is…

NKO!

The #1 Game

PLIN

Plinko is played so ofte
available:

2000-2011: pla
Total ch

Total chips in $10,00
Total winnings: $2,214

Average winning

e on TPIR is…

NKO!

en that great data is

ayed 308 times!
hips: 1,227
00 space: 176 (14.3%)

4,600 ($1,805 per chip)
gs per play: $7,190

Backtrack

How much is this Plink

king Plinko

ko chip worth right now?

Backtracking Plinko

5000
100 500 1000 0 10000 0

Each entry
is the value
of a chip at
that spot.

We know
the last
row…

1000 500 100

Backtracking Plinko

300 750 500 5000 5000 5
100 500 1000 0 10000 0

Work from
the bottom
up… each
number is
the mean
of the two
below it!

500 750 300

1000 500 100

Backtracking Plinko 300 525 625 2750 5000 2750
300 750 500 5000 5000 5

100 500 1000 0 10000 0

Work from
the bottom
up… each
number is
the mean
of the two
below it!

0 625 525 300
500 750 300

1000 500 100

Backtracking Plinko 413 575 1688 3875 3875 1
300 525 625 2750 5000 2750

300 750 500 5000 5000 5
100 500 1000 0 10000 0

1688 575 413 Work from
0 625 525 300 the bottom
500 750 300 up… each
number is
1000 500 100 the mean
of the two
below it!

Backtracking Plinko 413 494 1131 2781 3875 2781
413 575 1688 3875 3875 1

300 525 625 2750 5000 2750
300 750 500 5000 5000 5

100 500 1000 0 10000 0

1 1131 494 413 Work from
1688 575 413 the bottom
0 625 525 300 up… each
500 750 300 number is
the mean
1000 500 100 of the two
below it!

Backtracking Plinko 453 813 1956 3328 3328 1
413 494 1131 2781 3875 2781

413 575 1688 3875 3875 1
300 525 625 2750 5000 2750

300 750 500 5000 5000 5
100 500 1000 0 10000 0

1956 813 453 Work from
1 1131 494 413 the bottom
1688 575 413 up… each
0 625 525 300 number is
500 750 300 the mean
of the two
1000 500 100 below it!

Backtracking Plinko 453 633 1384 2642 3328 2642
453 813 1956 3328 3328 1

413 494 1131 2781 3875 2781
413 575 1688 3875 3875 1

300 525 625 2750 5000 2750
300 750 500 5000 5000 5

100 500 1000 0 10000 0

In the long
run, it all
evens out.

2 1384 633 453 But this
1956 813 453 isn’t a long
1 1131 494 413 run…
1688 575 413
0 625 525 300
500 750 300

1000 500 100

Backtracking Plinko 543 1009 2013 2985 2985 2
453 633 1384 2642 3328 2642

453 813 1956 3328 3328 1
413 494 1131 2781 3875 2781

413 575 1688 3875 3875 1
300 525 625 2750 5000 2750

300 750 500 5000 5000 5
100 500 1000 0 10000 0

2013 1009 543 In the long
2 1384 633 453 run, it all
1956 813 453 evens out.
1 1131 494 413
1688 575 413 But this
0 625 525 300 isn’t a long
500 750 300 run…

1000 500 100

Backtracking Plinko 543 776 1511 2499 2985 2499
543 1009 2013 2985 2985 2

453 633 1384 2642 3328 2642
453 813 1956 3328 3328 1

413 494 1131 2781 3875 2781
413 575 1688 3875 3875 1

300 525 625 2750 5000 2750
300 750 500 5000 5000 5

100 500 1000 0 10000 0

9 1511 776 543 In the long
2013 1009 543 run, it all
2 1384 633 453 evens out.
1956 813 453
1 1131 494 413 But this
1688 575 413 isn’t a long
0 625 525 300 run…
500 750 300

1000 500 100

Backtracking Plinko 659 1143 2005 2742 2742 2
543 776 1511 2499 2985 2499

543 1009 2013 2985 2985 2
453 633 1384 2642 3328 2642

453 813 1956 3328 3328 1
413 494 1131 2781 3875 2781

413 575 1688 3875 3875 1
300 525 625 2750 5000 2750

300 750 500 5000 5000 5
100 500 1000 0 10000 0

2005 1143 659 In the long
9 1511 776 543 run, it all
2013 1009 543 evens out.
2 1384 633 453
1956 813 453 But this
1 1131 494 413 isn’t a long
1688 575 413 run…
0 625 525 300
500 750 300

1000 500 100

Backtracking Plinko 659 901 1574 2374 2742 2374
659 1143 2005 2742 2742 2

543 776 1511 2499 2985 2499
543 1009 2013 2985 2985 2

453 633 1384 2642 3328 2642
453 813 1956 3328 3328 1

413 494 1131 2781 3875 2781
413 575 1688 3875 3875 1

300 525 625 2750 5000 2750
300 750 500 5000 5000 5

100 500 1000 0 10000 0

4 1574 901 659 In the long
2005 1143 659 run, it all
9 1511 776 543 evens out.
2013 1009 543
2 1384 633 453 But this
1956 813 453 isn’t a long
1 1131 494 413 run…
1688 575 413
0 625 525 300
500 750 300

1000 500 100

Backtracking Plinko 780 1238 1974 2558 2558 1
659 901 1574 2374 2742 2374

659 1143 2005 2742 2742 2
543 776 1511 2499 2985 2499

543 1009 2013 2985 2985 2
453 633 1384 2642 3328 2642

453 813 1956 3328 3328 1
413 494 1131 2781 3875 2781

413 575 1688 3875 3875 1
300 525 625 2750 5000 2750

300 750 500 5000 5000 5
100 500 1000 0 10000 0

1974 1238 780 At the top
4 1574 901 659 we find the
2005 1143 659 expected
9 1511 776 543 value for
2013 1009 543 dropping a
2 1384 633 453 chip from
1956 813 453 each slot!
1 1131 494 413
1688 575 413
0 625 525 300
500 750 300

1000 500 100

Backtracking Plinko 780 1009 1606 2266 2558 2266
780 1238 1974 2558 2558 1

659 901 1574 2374 2742 2374
659 1143 2005 2742 2742 2

543 776 1511 2499 2985 2499
543 1009 2013 2985 2985 2

453 633 1384 2642 3328 2642
453 813 1956 3328 3328 1

413 494 1131 2781 3875 2781
413 575 1688 3875 3875 1

300 525 625 2750 5000 2750
300 750 500 5000 5000 5

100 500 1000 0 10000 0

6 1606 1009 780 At the top
1974 1238 780 we find the
4 1574 901 659 expected
2005 1143 659 value for
9 1511 776 543 dropping a
2013 1009 543 chip from
2 1384 633 453 each slot!
1956 813 453
1 1131 494 413
1688 575 413
0 625 525 300
500 750 300

1000 500 100

Good Plin

Where you drop Plinko

Drop Above Chip EV

$10,000
$0

$1,000
$500
$100

nko Advice

o chips matters a lot!

Good Plin

Where you drop Plinko

Drop Above Chip EV

$10,000 $2,558
$0 $2,266
$1,606
$1,000 $1,009
$500 $780
$100

nko Advice

o chips matters a lot!