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Suppose that a basketball player's success in free-throw shooting can be described with a Markov chain with transition matrix
The probabilities are if the player makes a free throw, then he will make the next one with a probability of .4, and he will miss it with a probability of .6. If the player misses a free throw, then he will make the next one with a probability of .7 and miss it with a probability of .3.
If the player misses his first free throw, what is the probability that he also misses the third one?
If the player misses his first free throw, what is the probability that he makes the thrid one?
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The transition matrix is `P=([.4,.6],[.7,.3])` (Assuming that state 1 is making a shot, and state 2 is missing a shot.) `p_(1,1)` is the entry in row 1 column 1 and is the probability of going from state 1 to state 1.
Then we are interested in `P^3` , the third repetition of the experiment.
(1) If he misses the first shot (state 2) and misses the third shot (also state 2) then we look at `p_(2,2)=.447` to find the probability.
The probability that he misses the first and third shot is .447
(2) If he misses the first shot (state 2) and makes the third shot (state 1) we look at `p(2,1)=.553`
The probability that he misses the first shot and makes the third shot is .553
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