Sets

01cdb4ac-35e3-41d4-9b52-7d9cb4c7503e 50 4 5 false false 0 decimal

&[DATE] &[TIME] - &[FILENAME]

Sets

Binary set operations X O Y S O ∈ ≡ R X Ø ≡ (Y transpose X 1 findrows 0 ≠ (| : O ⊂ ≡ R X ⊆ Y S X Y ≠ (& (: O ⊆ ≡ R X ∩ Y S X ≡ (: R Ø : k 0 : n Y : 13 mat O RemDup ≡ n 2 < R X : Q X 1 csort : R 1 el Q 1 el : k 1 : 13 mat ν 2 n range Q ν el Q ν 1 - el ≡ continue R 1 k k 1 + : el Q ν el : if for 21 line if O \00AC\ ≡ X Ø ≡ (Y Ø ≡ (| R Ø : y Y y ∈ X S 0 ≡ R 1 k k 1 + : el y : continue if for if O ∪ ≡ X Ø ≡ R Y : Y Ø ≡ R X : x X R 1 k k 1 + : el x : for y Y R 1 k k 1 + : el y : for R RemDup k 1 - S 31 line if if O ∩ ≡ x X y Y x y ≡ R 1 k k 1 + : el x : continue if for for O × ≡ x X y Y R 1 k k 1 + : el x y 12 mat : for for Bad Operator error if if if if if 21 line if if if R 21 line :

Set constructor X S R Ø : Q R : n 0 : 13 mat x X Û transpose x 1 findrows 0 ≠ Q n n 1 + : el x : 0 if for Q RemDup n S 31 line :

Unary set operations O X S O \00AC\ ≡ X Û ≡ Ø X \00AC\ Û S if Bad Operator error if 11 line :

Empty set Ø 011 mat 11 line :

It could be also

Ø Û max 11 mat :

Cardinal function. X L X length 11 line :

Returns the indices where the elements of X are one. X Find.1 k 1 X L range : I 0 : 12 mat I k el X k el : k augment eval 11 findrows 2 col 21 line :

Stem plot of X, Y data. X Y PlotStem N X L : S 0 : k 1 N range : clr 26 range : 14 mat k 1 N range S 6 k * 1 + el X k el Y k el X k el 022 mat : S 6 k 1 - (* clr + el : 12 mat for S 1 el X 1 N range el Y 1 N range el o 10 blue augment : mat2sys.1 31 line :

X Plot2Die Πo 0 : x 0 : y 0 : F ID2 φ : r 1 F L range : 15 mat Πo x r el 0.1 F r el 1 el + : y r el 0.1 F r el 2 el + : ■ 20 green augment : Πo X 1 X L range el 5 el red : 31 line :

X product X κ el κ κ Clear X L product 11 line :

Alvaro

Exercise

Conditional Probability

This exercise shows that probabilities are a measuring theory, and the importance of make experiments to have the actual values for physical or engieneering events, instead of the simple assignation of theoretical probability values for those events.

The universe of events for a die is

D One Two Three Four Five Six 16 mat :

indexed by

ID 1 D L range transpose :

with empty set

Empty set Ø 011 mat 11 line :

Thus, the row vector

C 13313 mat :

represents the event for get One, then Three and then Three again

D 1 C el One Three Three 13 mat

The probability for an event given by a row vector X is

P Clear 1

X P.D P X el product 0 try 11 line :

where P is the probability of each die number. For a perfect die we have

P 16 / 16 / 16 / 16 / 16 / 16 / 16 mat :

ID P PlotStem

then the probability for C is

C P.D 1216 /

16 / 16 / * 16 / * 1216 /

this is

Now, if rolling the same die twice, we have

D2 D × D S : One One 12 mat One Two 12 mat 12 mat

and the indices are

ID2 1 D2 L range transpose : 123456789101112112 mat

For correlate both universes define φ as

ID2 ID × ID S :

X φ ID2 1 X el :

For example, the row vector C

C 7281213 mat :

C Plot2Die

maped by

C φ 2112 mat 5412 mat 2612 mat 13 mat

represents the event of get Two and One, or Five and Four, or Two and Six

D2 1 C el Two One 12 mat Five Four 12 mat Two Six 12 mat 13 mat

The probability in the 'rolling die twice' space is

P.2S P × P S 11 line :

X P.2S P.2S X el κ el product κ 1 X L sum 0 try 11 line :

ID2 P.2S 1

It's well defined because in the universe 'ID2 because for each X⊆ID2 is

Ø P.2S 0

X P.2S 01 lele

For the event given by C, it is

C P.2S 112 /

336 / 112 /

Defining an event A by 'first roll is two', we have

X f.A X φ 1 el 2 ≡ :

A ID2 f.A vectorize Find.1 transpose : 78910111216 mat

A Plot2Die

A P.2S 16 /

636 / 16 /

D2 1 A el Two One 12 mat Two Two 12 mat Two Three 12 mat 13 mat

Defining an event B by 'the sum for two rolls is less or equal than five', we have

X f.B X φ 1 el X φ 2 el + 5 ≤ :

B ID2 f.B vectorize Find.1 transpose : 12347816 mat

B Plot2Die

B P.2S 518 /

1036 / 518 /

D2 1 B el One One 12 mat One Two 12 mat One Three 12 mat 13 mat

The event given by the intersection of the two previous cases is

A ∩ B S P.2S 112 /

336 / 112 /

A ∩ B S Plot2Die

D2 1 A ∩ B S el Two One 12 mat Two Two 12 mat Two Three 12 mat 13 mat

Finally, the conditional probability P(B|A) is

P.B\007C\A A ∩ B S P.2S B P.2S / : 310 /

That's ok for a perfect die. What about a real one? It's easy to make one, with a small attached mass, for example, in the six face. Rolling it 40 times, I found this events

E\0024\ Two Three Three Five Four Three Five One Two Three Five Two Five One Three Five Four Six Two Three Four Two One Four Five One Two Four Three Four Three Two One Two Four Three Four Six Three Two 410 mat :

The probability for each number is now

N E\0024\ L : 40

P 1 ID el E\0024\ 1 N range el D ID el 1 findrows L N / :

ID P PlotStem

P 0.125 0.225 0.25 0.2 0.15 0.05 16 mat

with

P sum 1

For this imperfect die now we have:

A P.2S 0.225

instead

636 / 0.1667

B P.2S 0.3475

1036 / 0.2778

336 / 0.0833

A ∩ B S P.2S 0.135

P.B\007C\A A ∩ B S P.2S B P.2S / : 0.3885

310 / 0.3

Exercise

Here, we roll the same die twice. Make a second imperfect die, and eval the conditional probability for the same events, but rolling each die once.

Rolling my second imperfect die, with a small mass at the face of Five, 30 times, I have

E\0024\.2 Six One Two Two Three Six Two Five One Four Four Two Four Three Four One Two Five Two Six Three One Two Four Two Two Six Four Six Two 310 mat :

Demostrante using SMath that now it is

P.B\007C\A 153440 / ≡

References

https://en.wikipedia.org/wiki/Conditional_probability#Example

Alvaro

Solution

Exercise

Here, we roll the same die twice. Make a second imperfect die, and eval the conditional probability for the same events, but rolling each die once.

First Die (the same as before)

E\0024\.1 Two Three Three Five Four Three Five One Two Three Five Two Five One Three Five Four Six Two Three Four Two One Four Five One Two Four Three Four Three Two One Two Four Three Four Six Three Two 410 mat :

Second Die (rolling it only 30 times)

E\0024\.2 Six One Two Two Three Six Two Five One Four Four Two Four Three Four One Two Five Two Six Three One Two Four Two Two Six Four Six Two 310 mat :

Probabilities for each die

E\0024\ P.Dn P 0 : P 1 ID el E\0024\ 1 E\0024\ L range el D ID el 1 findrows L E\0024\ L / : 21 line :

P.1 E\0024\.1 P.Dn :

ID P.1 PlotStem

P.2 E\0024\.2 P.Dn :

ID P.2 PlotStem

Now we have

P.2S P.1 × P.2 S 11 line :

and

A P.2S 0.225

insetad

636 / 0.1667

B P.2S 0.3667

1036 / 0.2778

336 / 0.0833

A ∩ B S P.2S 0.1275

P.B\007C\A A ∩ B S P.2S B P.2S / : 0.3477

310 / 0.3

Check: by hand

A φ 2112 mat 2212 mat 2312 mat 2412 mat 2512 mat 2612 mat 16 mat

P.2SA P.1 2 el P.2 1 el * P.1 2 el P.2 2 el * P.1 2 el P.2 3 el * P.1 2 el P.2 4 el * P.1 2 el P.2 5 el * P.1 2 el P.2 6 el * 16 mat sum : 0.225

B φ 1112 mat 1212 mat 1312 mat 1412 mat 2112 mat 2212 mat 2312 mat 3112 mat 3212 mat 4112 mat 110 mat

P.2SB P.1 1 el P.2 1 el * P.1 1 el P.2 2 el * P.1 1 el P.2 3 el * P.1 1 el P.2 4 el * P.1 2 el P.2 1 el * P.1 2 el P.2 2 el * P.1 2 el P.2 3 el * P.1 3 el P.2 1 el * P.1 3 el P.2 2 el * P.1 4 el P.2 1 el * 25 mat sum : 0.3667

A ∩ B S φ 2112 mat 2212 mat 2312 mat 13 mat

P.2S∩ P.1 2 el P.2 1 el * P.1 2 el P.2 2 el * P.1 2 el P.2 3 el * 13 mat sum : 0.1275

P.B\007C\A P.2S∩ P.2SB / : 153440 /

equals to

A ∩ B S P.2S B P.2S / 153440 /

Alvaro