Classroom Investigations of Recent Research Concerning the Hot Hand Phenomenon

Figures & data

Table 1. Results for Stephen Curry in the first round of the 2016 NBA three-point contest.

Attempt #	1	2	3	4	5	6	7	8	9	10	11	12	13
Outcome	0	0	0	1	1	0	1	0	1	1	1	1	1
Attempt #	14	15	16	17	18	19	20	21	22	23	24	25
Outcome	1	1	1	1	1	1	0	0	0	0	1	1

Table 2. Observed value of streak statistics, with a streak length of 3, for data in Table 1.

1	Proportion of S after streaks of S	8/9 = 0.8889
2	Difference in proportion of S (after streaks of S – other trials)	8/9 – 8/16 = 0.3889
3	Difference in proportion of S (after streaks of S – after streaks of F)	8/9 – 2/3 = 0.2222
4	Frequency of S streaks	9/22 = 0.4091
5	Longest run of S	11
6	Total number of runs	8

Figure 1. One hundred simulated values of streak statistic 1 with streak length 3 when there are 16 successes out of 25 trials.

Figure 2. Simulated null distribution of streak statistic 1 with streak length 3 when there are 16 successes in 25 trials.

Figure 3. Simulated null distribution of streak statistic 2 with streak length 3 when there are 16 successes in 25 trials.

Figure 4. Null distribution of streak statistic 1 with streak length 3 when there are 50 successes in 100 trials.

Figure 5. Simulated p-value based on the null distribution in Figure 2 and an observed value of 0.8889.

Figure 6. Values of streak statistic 2 with streak length 3 for the 37 players in Appendix A.

Figure 7. Values of z-score of streak statistic 2 with streak length 3 standardized with respect to the null distribution for the 37 players in Appendix A.

Figure 8. Values of the streak statistic and the bias-corrected values for the data in Appendix B.

Table 3. The sample space for Problem 1. Flips that follow H are in boldface.

Outcome	Number of flips following H (Z)	Number of flips following H that result in H (Y)	Proportion of flips following H that result in H (X = Y/Z)
HHHH	3	3	1
HHHT	3	2	2/3
HHTH	2	1	1/2
HTHH	2	1	1/2
THHH	2	2	1
HHTT	2	1	1/2
HTHT	2	0	0
HTTH	1	0	0
THHT	2	1	1/2
THTH	1	0	0
TTHH	1	1	1
HTTT	1	0	0
THTT	1	0	0
TTHT	1	0	0
TTTH	These outcomes are discarded.
TTTT

Table 4. The probability mass function of X, the proportion of flips following H that result in H in Problem 1.

X	0	1/2	2/3	1
P(X = x)	6/14	4/14	1/14	3/14

Table 5. The joint probability mass function, $p_{Y,Z}\left(y,z\right)=P\left(Y=y,Z=z\right)$, of Z, the number of flips immediately following H, and Y, the number of flips immediately following H that result in H, in Problem 1.

		z
		1	2	3
Y	0	5/14	1/14	0
	1	1/14	4/14	0
	2	0	1/14	1/14
	3	0	0	1/14

Table 6. Comparison of two-stage random processes.

	Problem 2 (restating Problem 1)	Problem 3
Stage 1	Flip a fair coin four times and record the results in sequence. If the outcome is TTTH or TTTT discard it and try again.	Flip a fair coin. If it lands T discard the trial and try again.
Stage 2	From the sequence observed in stage 1, select uniformly at random one of the flips which immediately follows an H, and record the result of the selected flip.	Flip the coin again and record the result.
P (Stage 2 results in H)	17/42	1/2