Solutions Homework 7, Math10170, Spring 2015
Question 1 The following table show the split statistics from the ESPN website for Basketball player
Stephen Curry for the 2014-2015 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)?
Using his career FT% as an estimate, we get the estimate P (F T ) ≈ 0.902.
(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)?
Using his career 3P% as an estimate, we get the estimate P (3P ) ≈ 0.4.
(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?
Using the above estimate P (F T ) ≈ 0.902 and independence we get
P (makes all 3) = P (makes 1st ∩ makes 2nd ∩ makes 3rd)
= P (makes 1st)P (makes 2nd)P (makes 3rd) = (0.902)3 ≈ 0.734.
(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
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point field goal attempt, estimate the probability that he will make all three of his next three point field
goal attempts ?
Using the above estimate P (3P ) ≈ 0.902 and independence we get
P (makes all 3) = P (makes 1st ∩ makes 2nd ∩ makes 3rd)
= P (makes 1st)P (makes 2nd)P (makes 3rd) = (0.4)3 ≈ 0.064.
(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?
Using the split statistics for Free throws made (out of FT attempted) after 3+ days rest, we get
P (F T |3 + days rest) ≈ 0.944. The question about whether the two events are independent can be
translated to the following question:
“Is P (F T |3 + days rest) = P (F T )?”
or “Is 0.944 = 0.904?
Clearly not, so we can conclude that the two events are not independent, however given more powerful
statistical tools, we would take a margin of error into account.
(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?
Using the split statistics for 3 PT FG’s made (out of 3 PT FG’s attempted) after 3+ days rest, we
get P (3P |3 + days rest) ≈ 0.351. The question about whether the two events are independent can be
translated to the following question:
“Is P (3P |3 + days rest) = P (3P )?”
or “Is 0.351 = 0.4?
Obviously not, so we can conclude that the two events are not independent, however given more powerful
statistical tools, we would take a margin of error into account.
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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
1
1
H
× K2
4
2
HH
3
..
.
N
HHH
..
.
HH
. . . H}
| {z
1
× K2
8
1
× K2
16
..
.
1
2N +1
×
K
2
N times
..
.
..
.
..
.
The longest run of heads in K flips of a coin: If a coin is flipped K times, we would expect the
ln( K )
length of the longest run to be around log2 ( K2 ) = ln 22 or the largest value of L for which 21L K2 rounds
to a whole number bigger than 0.
(a) If you flip a coin 100 times, how many runs of heads of length 1 (H), 2 (HH), 3 (HHH), 4 (HHHH),
5(HHHHH), should we expect?
We expect:
Length of run of Heads
1
Outcome
H
2
HH
3
HHH
4
HHHH
5
HHHHH
Expected Number
50
≈ 12
4
50
8
50
16
50
32
50
64
≈7
≈3
≈2
≈1
(b) If you flip a coin 100 times, what is the approximate length of the longest run of heads that we
should expect?
We expect the longest run to have length roughly equal to
ln 100
2
ln(50)
=
≈ 5.64 ≈ 6.
ln(2)
ln(2)
(c) Find the number of runs of heads of the given lengths in the following sequences of heads and tails:
Sequence 1:
TTTTHTTTHTTHHHTHTTTHHHHTTHTHTTHHHTTHTTTTTTT
HHTTHHHHHTHTTTHHTTTHHTHTTTHHTHTTHTTHTTTTHHT
HHHHHTTHTTHTTT
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Sequence 2:
HTHTTHTTHHTHTHHTTHHTTHTTTHHTHTHTTHTHTTHTTTH
THHTHHTHTTHHTHTHHHTTHTTTHTTHTTHHTHTTHHHTTTT
HHHTHTTHHTHHTT
Length of run of Heads
1
Outcome
H
Number in Seq. 1
13
Number in Seq. 2
18
2
HH
5
10
3
HHH
2
3
4
HHHH
1
0
5
HHHHH
2
0
6
HHHHHH
0
0
7
HHHHHHH
0
0
(d) What is the length of the longest run of heads in both sequences?
In sequence 1, the longest run of heads has length 5 and in sequence 2, the longest run of heads has
length 3.
(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.
Sequence 2 was the made up sequence. We expect the longest run in a sequence of 100 flips of a coin
to have length roughly
ln(50)
≈ 6.
ln(2)
Because the distribution of runs in sequence 1 is a better match for the number of runs we would expect
in data generated randomly, we claim that it was the randomly generated sequence.
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Question 3 We saw in class that the FG% for LeBron James was approximately 0.5. The following set
of data shows whether he got a basket or a miss on a string of 319 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.
1
B
42
2
BB
26
3
BBB
8
4
BBBB
4
5
BBBBB
3
6
BBBBBB
0
7
BBBBBBB
1
8
BBBBBBBB
0
(b) What is the longest run of baskets in the above sequence of shots?
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(c) If the sequence of shots was randomly generated with a probability of a basket equal to 0.5 on each
shot, what would you expect for the length of the longest run?
ln 319
2
ln(159.5)
=
≈ 7.3 ≈ 7.
ln(2)
ln(2)
(d) Based on your observations would you consider LeBron James to be a streaky player? No, this
string of baskets and misses fits the profile of data generated randomly with a 50% chance of success
on each shot.
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Question 4 If a professional basketball player played in 800 games throughout his career making an
average of 10 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?
Throughout the course of his career, this player would take about 800 × 10 = 8, 000 consecutive
shots. We would expect the longest run to have length approximately
ln(4000)
≈ 11.96.
ln(2)
Thus we would expect the longest run of consecutive field goals to be about 12 shots in length.
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