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A monkey randomly types letters fA; B; C; D; E; F; G; H; I; J; K; L; M; N; O; P; Q; R; S; T; U; V; W; X; Y; Zg. What
is the expected number of key presses until ABRACADABRA is generated?
Solution
The transition matrix for this markov chain is given by
0
25
1
0
0
0
0
26
26
B 24
1
1
B 26 26
0
0
0
26
B
1
B 24 1
0
0
0
B 26 26
26
B 25
1
0
0 26
0
B 26 0
B 23 1
1
1
B
0
0
26
B 26 26 26
B 25 0
0
0
0
0
B
P D B 26
1
1
B 23
0
0
0
B 26 26 26
B 25 0
0
0
0
0
B 26
B 24 1
0
0
0
B 26 26 0
B 24 1
B
0
0
0
0
B 26 26
B 25 0
0
0
0
0
@ 26
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
26
0
0
0
0
0
0
0
0
0
0
0
0
0
1
26
0
0
0
0
0
0
0
0
1
26
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
26
0
0
0
0
1
26
0
0
0
0
0
A diagram for this system can be seen in the following figure.
¿
abracadabr
abracadab
a
a
r
a
b
r
b
ab
abracada
a
abracad
abr
abraca
d
abra
abrac
a
The absorption equations are described by
i D 1 C
m
X
pij j :
j D1
This leads to a system of linear equations in 1 ; ; 12 .
1 of 2
c
a
0
0
0
0
0
0
0
0
0
0
1
26
1
1
C
C
C
C
C
C
C
C
C
C
C
C
C:
C
C
C
C
C
C
C
C
C
C
A
8
1
ˆ
C 2 26
1 D 1 C 1 25
ˆ
26
ˆ
ˆ
ˆ
1
1
ˆ
2 D 1 C 1 24
C 2 26
C 3 26
ˆ
26
ˆ
ˆ
24
1
1
ˆ
ˆ
3 D 1 C 1 26 C 2 26 C 4 26
ˆ
ˆ
ˆ
1
ˆ
4 D 1 C 1 25
C 5 26
ˆ
ˆ
26
ˆ
ˆ
23
1
1
ˆ
5 D 1 C 1 26 C 2 26
C 3 26
C 6
ˆ
ˆ
ˆ
< D 1 C 25 C 1
7 26
6
1 26
1
1
ˆ
7 D 1 C 1 23
C 2 26
C 3 26
C 8
ˆ
26
ˆ
ˆ
25
1
ˆ
ˆ
D
1
C
C
8
1 26
9 26
ˆ
ˆ
ˆ
24
1
1
ˆ
D
1
C
C
ˆ
9
1
2 26 C 10 26
ˆ
26
ˆ
ˆ
24
1
1
ˆ
10 D 1 C 1 26 C 2 26 C 11 26
ˆ
ˆ
ˆ
ˆ
ˆ11 D 1 C 1 25 C 12 1
ˆ
26
26
ˆ
ˆ
: D 0
12
Represented in matrix form:
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
26
24
26
24
26
25
26
23
26
25
26
23
26
25
26
24
26
24
26
25
26
0
1
26
25
26
1
26
0
1
26
0
1
0
1
26
1
26
0
0
1
26
1
26
0
0
0
0
0
0
1
26
1
26
0
0
This matrix row Reduces to
0
1 0 0
B
B0 1 0
B
B0 0 1
B
B0 0 0
B
B0 0 0
B
B0 0 0
B
B0 0 0
B
B0 0 0
B
B0 0 0
B
B0 0 0
B
@0 0 0
0 0 0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
26
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
26
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
26
1
26
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
26
1
26
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
26
1
0
0
1
26
1
26
0
0
0
0
0
0
0
0
0
1
26
1
0
0
0
0
0
0
0
0
0
0
0
1
26
1
3670344487444778
3670344487444752
3670344487444102
3670344487427202
3670344486987776
3670344475563402
3670344178528976
3670336455634602
3670135660380176
3664914983765126
3529177391773826
0
Therefore, if one key is pressed per second, the expected time to ABRACADABRA is
116; 306; 000 years:
2 of 2
1
1
1
1
1
1
1
1
1
1
1
0
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
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