Question 2 We saw in class that if we flip a coin K times, the the distribution in runs of success in the resulting sequence is equal to

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1 Homework 7, Math070, Spring 205 Question The following table show the split statistics from the ESPN website for Basketball player Stephen Curry for the season so far (a) Using relative frequency as a measure of probability, what is your estimate of the probability that Stephen Curry will make the next free throw he takes ( if you have no information about other factors which might influence the probability)? (b) Using relative frequency as a measure of probability, what is your estimate of the probability that Stephen Curry will make the next three point field goal he attempts ( if you have no information about other factors which might influence the probability)? (c) Using the relative frequency as a measure of probability and assuming that Stephen Curry s performance on each free throw is independent of how he performs on any other free throw, estimate the probability that he will make all three of his next three free throws?

2 (d) Using the relative frequency as a measure of probability and assuming that Stephen Curry s performance on each three point field goal attempt is independent of how he performs on any other three point field goal attempt, estimate the probability that he will make all three of his next three point field goal attempts? (e) Use the statistics given to estimate the conditional probability that Stephen Curry will make his next attempted free throw given that he has had a three + day rest Are the events being successful on a free throw attempt and having a 3+ day rest independent for Stephen Curry? (f) Use the statistics given to estimate the conditional probability that Stephen Curry will make his next attempted three point field goal given that he has had a three + day rest Are the events being successful on a three point field goal attempt and having a 3+ day rest independent for Stephen Curry? Question 2 We saw in class that if we flip a coin K times, the the distribution in runs of success in the resulting sequence is equal to Experiment: Flip a coin K times Length of run of Heads Outcome Expected Number H 4 K 2 2 HH 8 K 2 3 HHH 6 K 2 N HH }{{ H} N times 2 N+ K 2 The longest run of heads in K flips of a coin: If a coin is flipped K times, we would expect the length of the longest run to be around log 2 ( K ) = ln( K 2 ) ( or the largest value of L for which K ) 2 ln 2 2 L 2 rounds to a whole number bigger than 0 2

3 (a) If you flip a coin 00 times, how many runs of heads of length (H), 2 (HH), 3 (HHH), 4 (HHHH), 5(HHHHH), should we expect? (b) If you flip a coin 00 times, what is the approximate length of the longest run of heads that we should expect? (c) Find the number of runs of heads of the given lengths in the following sequences of heads and tails: Sequence : T T T T H T T T H T T H H H T H T T T H H H H T T H T H T T H H H T T H T T T T T T T H H T T H H H H H T H T T T H H T T T H H T H T T T H H T H T T H T T H T T T T H H T H H H H H T T H T T H T T T Sequence 2: H T H T T H T T H H T H T H H T T H H T T H T T T H H T H T H T T H T H T T H T T T H T H H T H H T H T T H H T H T H H H T T H T T T H T T H T T H H T H T T H H H T T T T H H H T H T T H H T H H T T Length of run of Heads Outcome Number in Seq Number in Seq 2 H 2 HH 3 HHH 4 HHHH 5 HHHHH 6 HHHHHH 7 HHHHHHH (d) What is the length of the longest run of heads in both sequences? (e) One of the above sequences was generated by actually flipping a coin and the other was made up Can you tell which is which? Give reasons for your answer 3

4 Question 3 We saw in class that the FG% for LeBron James was approximately 05 The following set of data shows whether he got a basket or a miss on a string of 39 consecutive field goal attempts BBMMBMBBMMMMMBBBMBMMBMBMMB MBBMBBMMMMMBBMMM BBBMMMBBBMBMMMMB BBBMMMBBMMBMMMBBBMBMM MMBMBMBBBBBMMB BMBBMMBBMMMMMMMBM BMMBBMBBMMBBMBMBMMBBBBBMM BMMMBMMMBMMBBBBM MBMMMBBBBMBMMBBMBMMMMBBM MBMMMBBMBBMMMBMBMBBBMMMMB MMBBBBBBBMBBMMM BMMBMBBBMMBMBMMB BMBMMBBMMMBBMMBBBMBMBMB MBBMBBMMBMBMMMBBMMBMBBM MBMMBBBMBMMBBBBMBMMBMMMB MBMBBMMBBMBBBBBMBM (a) Count the number of runs of baskets of each length in the data and also number of each type that one would expect in data generated randomly with a 50% chance of success on each trial Length of run of Baskets Outcome Number in Seq Number in Seq 2 B 2 BB 3 BBB 4 BBBB 5 BBBBB 6 BBBBBB 7 BBBBBBB 8 BBBBBBBB (b) What is the longest run of baskets in the above sequence of shots? (c) If the sequence of shots was randomly generated with a probability of a basket equal to 05 on each shot, what would you expect for the length of the longest run? (d) Based on your observations would you consider LeBron James to be a streaky player? 4

5 Question 4 If a professional basketball player played in 800 games throughout his career making an average of 0 field goal attempts per game, with a 50% chance of getting a basket on each, what is the longest run of consecutive baskets (from field goal attempts) you would expect to see in the records for that player? 5