Tommy the dunker's performance on the basketball court is influenced by his state of mind. If he scores, he is three times as likely to score on the next shot as he is to miss; whereas, if he misses a shot he is four times as likely to miss the next shot as he is to score. In the long term, what percentage of shots are successful
A) $\frac { 4 } { 81 }$
B) $\frac { 5 } { 9 }$
C) $\frac { 2 } { 3 }$
D) $\frac { 4 } { 9 }$
E) $\frac { 6 } { 7 }$

Question

Tommy the dunker's performance on the basketball court is influenced by his state of mind. If he scores, he is three times as likely to score on the next shot as he is to miss; whereas, if he misses a shot he is four times as likely to miss the next shot as he is to score. In the long term, what percentage of shots are successful   
A) $\frac { 4 } { 81 }$
B) $\frac { 5 } { 9 }$
C) $\frac { 2 } { 3 }$
D) $\frac { 4 } { 9 }$
E) $\frac { 6 } { 7 }$

Quizplus · Accepted Answer

The problem can be modeled using a Markov chain with two states: scoring and missing. The transition probabilities are 3/4 for scoring after scoring and 1/5 for scoring after missing. Solving the steady-state probabilities gives the long-term probability of scoring as 4/9.

Tommy the Dunker's Performance on the Basketball Court Is Influenced 481\frac { 4 } { 81 }814​

Tommy the Dunker's Performance on the Basketball Court Is Influenced $\frac { 4 } { 81 }$