Gamblers A and B play a sequence of games defined as follows: A regular 16 faced die is rolled. . If the number is 8 or smaller, A wins; otherwise B wins. • Gambler A starts with $4, B starts with $3, and each wagers $1 in each game. The winner gets the $2 wagered in that game. The game ends when either gambler has $0. (a) What is the probability that gambler A is eventually ruined? (b) The first 4 state probability vectors are as follows: 0 1 0 TO = [0] 0 π1 = [0 0 0 0 [0 0 0.25 0.5 0 0.5 π2 0 0.5 0 T3 = [ 0 0.125 0 0.375 0 0.375 0 - 0 0 0.25 0 0 0 0.125] Given that after 3 games the game is in stateX3 = 5, what is the probability that the previous state was X₂ = 4?
Gamblers A and B play a sequence of games defined as follows: A regular 16 faced die is rolled. . If the number is 8 or smaller, A wins; otherwise B wins. • Gambler A starts with $4, B starts with $3, and each wagers $1 in each game. The winner gets the $2 wagered in that game. The game ends when either gambler has $0. (a) What is the probability that gambler A is eventually ruined? (b) The first 4 state probability vectors are as follows: 0 1 0 TO = [0] 0 π1 = [0 0 0 0 [0 0 0.25 0.5 0 0.5 π2 0 0.5 0 T3 = [ 0 0.125 0 0.375 0 0.375 0 - 0 0 0.25 0 0 0 0.125] Given that after 3 games the game is in stateX3 = 5, what is the probability that the previous state was X₂ = 4?
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![Problem 6
Gamblers A and B play a sequence of games defined as follows:
A regular 16 faced die is rolled.
. If the number is 8 or smaller, A wins; otherwise B wins.
. Gambler A starts with $4, B starts with $3, and each wagers $1 in each game.
The winner gets the $2 wagered in that game.
. The game ends when either gambler has $0.
(a) What is the probability that gambler A is eventually ruined?
(b) The first 4 state probability vectors are as follows:
TO
[0 0
[00
π1
π2
[0
0
T3 = [ 0 0.125
=
0
0
1
0
0.5
0 0.5
0
0.25
0 0.5
0
0 0.375 0 0.375 0
0
0
0
0
0.25 0
]
1
0.125]
Given that after 3 games the game is in stateX3 = 5, what is the probability that the previous state
was X₂ = 4?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc231fac6-f61a-45b2-b174-05bc9b27b74a%2F2fb6bdb6-03dd-4939-9554-8770a4b4952a%2Fhggjnol_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 6
Gamblers A and B play a sequence of games defined as follows:
A regular 16 faced die is rolled.
. If the number is 8 or smaller, A wins; otherwise B wins.
. Gambler A starts with $4, B starts with $3, and each wagers $1 in each game.
The winner gets the $2 wagered in that game.
. The game ends when either gambler has $0.
(a) What is the probability that gambler A is eventually ruined?
(b) The first 4 state probability vectors are as follows:
TO
[0 0
[00
π1
π2
[0
0
T3 = [ 0 0.125
=
0
0
1
0
0.5
0 0.5
0
0.25
0 0.5
0
0 0.375 0 0.375 0
0
0
0
0
0.25 0
]
1
0.125]
Given that after 3 games the game is in stateX3 = 5, what is the probability that the previous state
was X₂ = 4?
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