# Import the "random" string directly from Google
library(tidyverse)
library(googlesheets4)
gs4_deauth()
key <- "1wY3g7GSmii74oEOTtz2Id3B678jmfG_Ri4utO1jLn70"
ds <- read_sheet(key) |>
select(String, Length) |>
na.omit()Demo: Randomness
Human flips
First, we pull the contents of the Google spreadsheet into R. Make sure that you have curated the results a bit beforehand.
Next, we want to concatenate all of the T’s and H’s into one long string.
# Convert the random string into runs
human_flips <- ds |>
pull(String) |>
paste(collapse = "") |>
tolower() |>
str_split_1("")
human_flips [1] "h" "t" "h" "t" "h" "t" "h" "t" "h" "h" "h" "h" "h" "t" "t" "t" "t" "t"
[19] "h" "t" "h" "t" "h" "h" "t" "h" "t" "t" "t" "t" "t" "h" "h" "h" "t" "h"
[37] "h" "t" "h" "t" "h" "t" "t" "t" "h" "t" "h" "h" "t" "h" "h" "h" "t" "h"
[55] "h" "t" "t" "t" "h" "h" "t" "h" "t" "h" "h" "t" "h" "t" "h" "h" "h" "h"
[73] "t" "h" "h" "h" "t" "h" "t" "h" "h" "t" "h" "t" "h" "t" "t" "t" "t" "h"
[91] "t" "h" "t" "t" "t" "t" "t" "h" "t" "h" "t" "t" "t" "t" "h" "h" "h" "h"
[109] "t" "h" "h" "t" "h" "h" "t" "h" "h" "h" "t" "t" "h" "t" "h" "t" "h" "h"
[127] "t" "h" "h" "h" "t" "h" "h" "t" "t" "t" "h" "h" "t" "h" "h" "t" "h" "t"
[145] "h" "t" "h" "t" "h" "t" "h" "h" "t" "h" "h" "t" "h" "h" "t" "h" "t" "h"
[163] "h" "t" "h" "t" "h" "h" "h" "h" "t" "h" "h" "t" "h" "t" "h" "t" "h" "t"
[181] "h" "t" "h" "t" "t" "h" "t" "h" "t" "h" "h" "t" "t" "t" "t" "t" "h" "t"
[199] "h" "t" "h" "t" "h" "t" "h" "h" "t" "h" "t" "h" "h" "t" "t" "t" "t" "t"
[217] "t" "h" "t" "h" "t" "h" "t" "h" "t" "t" "h" "t" "h" "h" "t" "h" "t" "h"
[235] "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "t"
[253] "t" "t" "t" "t" "t" "t" "t" "t" "t" "t" "t" "t" "t" "t" "h" "h" "h" "h"
[271] "h" "h" "h" "h" "h" "t" "t" "t" "h" "t" "h" "t" "h" "t" "h" "t" "t" "t"
[289] "t" "t" "h" "t" "h" "h" "h" "h" "h" "t" "t" "h" "h" "t" "t" "t" "h" "h"
[307] "h" "h" "t" "h" "t" "h" "h" "h" "h" "h" "t" "t" "h" "h" "t" "h" "h" "t"
[325] "t" "h" "t" "h" "t" "h" "t" "t" "t" "t" "t" "t" "t" "t" "t" "t" "t" "t"
[343] "t" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "t" "t" "t" "h" "t" "h"
[361] "t" "h" "h" "h" "h" "h" "h" "t" "h" "h" "h" "h" "h" "h" "h" "h" "h" "t"
[379] "h" "h" "h" "h" "h" "t" "h" "h" "h" "h" "t" "t" "t" "t" "t" "t" "h" "h"
[397] "h" "h" "h" "h" "t" "t" "h" "t" "t" "t" "h" "t" "h" "t" "h" "t" "h" "t"
[415] "h" "t" "h" "h" "t" "h" "h" "h" "h" "h" "h" "h" "t" "h" "h" "h" "h" "t"
[433] "h" "t" "t" "t" "t" "t" "t" "t" "h" "h" "h" "h" "t" "h" "t" "h" "t" "t"
[451] "t" "t" "t" "t" "t" "t" "h" "h" "h" "h" "h" "h" "h" "h" "h" "t" "h" "h"
[469] "t" "t" "t" "t" "t" "h" "h" "h" "h" "h" "t" "h" "t" "t" "t" "t" "h" "h"
[487] "t" "h" "t" "h" "t" "h" "h" "t" "h" "t" "h" "t" "h" "t" "t" "h" "t" "h"
[505] "t" "t" "h" "t" "h" "t" "h" "h" "h" "t" "t" "h" "t" "h" "t" "h" "t" "h"
[523] "t" "h" "h" "t" "h" "t" "t" "h" "t" "h" "t" "h" "t" "t" "h" "t" "h" "t"
[541] "h" "h" "t" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t"
[559] "t" "h" "t" "h" "t" "h" "h" "t" "h" "t" "h" "h" "t" "h" "h" "h" "t" "h"
[577] "t" "h" "t" "h" "t" "h" "t" "t" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h"
[595] "t" "h" "t" "h" "h" "h" "t" "t" "h" "t" "h" "t" "h" "t" "h" "h" "h" "h"
[613] "t" "t" "t" "t" "h" "t" "h" "h" "h" "h" "t" "t" "t" "h" "t" "h" "h" "h"
[631] "t" "h" "h" "t" "h" "t" "t" "h" "t" "h" "h" "h" "h" "t" "h" "h" "h" "t"
[649] "h" "t" "t" "h" "h" "t" "h" "t" "h" "t" "t" "h" "t" "h" "t" "h" "t" "t"
[667] "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "h" "t" "h" "h" "t" "h" "t"
[685] "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t"
[703] "h" "t" "h" "t" "t" "h" "t" "h" "h" "t" "h" "h" "t" "h" "h" "h" "h" "h"
[721] "h" "h" "h" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h"
[739] "t" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "t" "t" "t"
[757] "t" "t" "t" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "t" "t" "t"
[775] "h" "t" "h" "t" "h" "t" "t" "h" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t"
[793] "h" "t" "h" "t" "h" "t" "h" "h" "t" "h" "t" "h" "t" "h" "h" "t" "h" "t"
[811] "t" "h" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "t" "h" "h" "t" "h" "t"
[829] "h" "t" "h" "t" "h" "t" "h" "t" "h" "h" "t" "h" "t" "h" "t" "h" "t" "h"
[847] "t" "h" "t" "h" "t" "t" "h" "h" "t" "t" "h" "t" "h" "t" "h" "t" "h" "t"
[865] "h" "t" "h" "t" "t" "h" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t"
[883] "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "t" "h" "h" "t"
[901] "h" "t" "h" "t" "t" "h" "h" "t" "h" "t" "t" "h" "t" "h" "t" "h" "h" "h"
[919] "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "t" "h" "h" "t" "h"
[937] "h" "t" "h" "t" "h" "t" "t" "h" "t" "h" "t" "h" "t" "t" "h" "t" "h" "h"
[955] "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h"
[973] "t" "t" "t" "t" "h" "t" "h" "t" "h" "h" "t" "h" "t" "h" "t" "t" "h" "t"
[991] "t" "t" "t" "h" "t" "h" "t" "h" "h" "t" "h" "h" "t" "h" "t" "h" "t" "h"
[1009] "h" "t" "h" "h" "t" "h" "t" "h" "h" "t" "t" "h" "t" "h" "t" "h" "t" "h"
[1027] "t" "h" "t" "h" "t" "t" "t" "h" "t" "h" "t" "h" "h" "t" "h" "h" "h" "t"
[1045] "h" "t" "h" "t" "h" "h" "t" "h" "t" "h" "t" "h" "t" "h" "h" "h" "t" "h"
[1063] "h" "h" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "t" "h" "t" "h" "t" "t"
[1081] "t" "h" "h" "t" "h" "t" "h" "h" "h" "h" "h" "h" "h" "h" "h" "t" "h" "t"
[1099] "h" "t" "h" "h" "t" "t" "t" "t" "t" "t" "t" "t" "t" "t" "h" "h" "h" "h"
[1117] "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "t" "h" "h" "h" "h"
[1135] "h" "h" "h" "h" "t" "h" "t" "h" "t" "h" "h" "t" "h" "t" "h" "t" "t" "h"
[1153] "t" "t" "t" "t" "t" "t" "t" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t"
[1171] "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "t" "h" "t" "h" "t" "h" "t" "h"
[1189] "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h"
[1207] "t" "h" "t" "t" "h" "h" "t" "t" "t" "h" "t" "h" "t" "h" "h" "h" "t" "h"
[1225] "t" "h" "t" "h" "h" "h" "t" "h" "t" "h" "t" "h" "t" "h" "h" "t" "h" "h"
[1243] "h" "h" "h" "h" "h" "h" "t" "h" "t" "h" "h" "t" "h" "t" "h" "h" "t" "h"
[1261] "h" "t" "h" "t" "t" "t" "h" "t" "h" "h" "t" "h" "h" "h" "t" "t" "t" "h"
[1279] "h" "h" "t" "h" "h" "h" "t" "h" "h" "h" "t" "t" "h" "h" "t" "h" "h" "t"
[1297] "h" "h" "h" "t" "t" "h" "h" "t" "h" "h" "t" "h" "h" "h" "t" "h" "t" "h"
[1315] "h" "h" "t" "h" "h" "h" "t" "h" "h" "h" "h" "t" "t" "h" "t" "t" "t" "t"
[1333] "h" "t" "t" "h" "h" "t" "t" "h" "t" "t" "h" "h" "t" "h" "t" "h" "h" "t"
[1351] "h" "t" "h" "h" "t" "h" "h" "h" "t" "h" "t" "t" "h" "h" "t" "h" "t" "h"
[1369] "h" "h" "t" "h" "t" "t" "h" "t" "t" "h" "t" "h" "h" "t" "h" "t" "h" "h"
[1387] "t" "h" "t" "h" "t" "h" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "h"
[1405] "t" "h" "t" "h" "h" "t" "h" "t" "h" "t" "h" "t" "h" "h" "t" "h" "t" "h"
[1423] "t" "h" "t" "h" "h" "t" "h" "t" "h" "t" "h" "t" "h" "h" "h" "t" "h" "t"
[1441] "h" "t" "h" "t" "h" "t" "h" "h" "t" "t" "h" "t" "h" "t" "h" "t" "h" "t"
[1459] "h" "t" "h" "t" "h" "h" "t" "t" "h" "t" "h" "t" "h" "h" "t" "h" "h" "t"
[1477] "t" "h" "t" "h" "t" "h" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t"
[1495] "h" "t" "h" "h" "t" "t" "h" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h"
[1513] "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "h" "t" "h" "h"
[1531] "h" "h" "t" "h" "t" "t" "t" "h" "h" "h" "t" "h" "h" "t" "h" "h" "t" "h"
[1549] "h" "t" "h" "t" "h" "h" "t" "h" "h" "t" "h" "h" "t" "h" "h" "h" "h" "h"
[1567] "h" "t" "t" "t" "t" "t" "t" "t" "t" "t" "t" "h" "t" "h" "h" "t" "h" "t"
[1585] "h" "t" "h" "t" "h" "h" "t" "h" "h" "t" "h" "h" "t" "h" "t" "h" "h" "t"
[1603] "h" "h" "t" "h" "t" "h" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t"
[1621] "h" "t" "h" "t" "h" "t" "t" "t" "t" "t" "t" "t" "t" "t" "h" "h" "h" "h"
[1639] "h" "h" "h" "h" "t" "h" "t" "t" "t" "t" "t" "h" "h" "h" "h" "h" "h" "h"
[1657] "t" "h" "t" "h" "t" "h" "t" "h" "h" "t" "t" "t" "t" "t" "h" "h" "h" "h"
[1675] "h" "h" "t" "t" "h" "t" "h" "h" "h" "h" "h" "h" "t" "t" "t" "h" "t" "h"
[1693] "h" "h" "h" "h" "t" "h" "t" "t" "t" "h" "h" "t" "h" "t" "h" "t" "h" "h"
[1711] "h" "t" "t" "t" "h" "h" "h" "h" "h" "h" "h" "h" "h" "t" "t" "h" "h" "h"
[1729] "h" "t" "t" "t" "t" "t" "h" "h" "h" "t" "t" "t" "h" "h" "t" "h" "h" "h"
[1747] "t" "h" "t" "h" "h" "t" "h" "t" "h" "t" "t" "t" "h" "t" "h" "t" "h" "t"
[1765] "h" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "h" "t"
[1783] "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t"
[1801] "h" "t" "h" "t" "h" "t" "h" "t" "h" "h" "t" "h" "t" "h" "t" "h" "t" "h"
[1819] "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "h" "t" "h" "t" "h" "t" "h" "t"
[1837] "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t"
[1855] "h" "t" "h" "h" "t" "h" "t" "h" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t"
[1873] "t" "h" "t" "h" "t" "h" "t" "h" "h" "t" "t" "t" "h" "h" "t" "h" "t" "h"
[1891] "t" "h" "t" "t" "h" "t" "h" "t" "h" "t" "h" "t" "t" "h" "t" "h" "h" "t"
[1909] "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "t" "h" "t" "h" "t" "t" "h"
[1927] "t" "t" "h" "h" "t" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t"
[1945] "t" "t" "h" "t" "t" "h" "t" "h" "t" "t" "t" "t" "t" "t" "t" "t" "t" "h"
[1963] "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "t" "t" "t" "t"
[1981] "t" "t" "t" "t" "t" "t" "t" "t" "h" "h" "h" "h" "h" "h" "t" "t" "h" "t"
[1999] "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "h" "h" "h"
[2017] "h" "h" "h" "t" "t" "t" "t" "t" "t" "h" "h" "h" "h" "h" "h" "h" "t" "t"
[2035] "t" "t" "t" "t" "t" "h" "h" "h" "h" "h" "h" "h" "h" "t" "h" "t" "h" "t"
[2053] "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t"
[2071] "h" "t" "h" "t" "h" "h" "h" "h" "h" "h" "h" "h" "t" "h" "h" "t" "h" "t"
[2089] "h" "t" "h" "t" "h" "t" "h" "t" "h" "h" "h" "h" "h" "h" "h" "h" "t" "t"
[2107] "t" "t" "t" "t" "t" "t" "t" "t" "h" "h" "h" "h" "h" "t" "t" "t" "t" "t"
[2125] "h" "h" "h" "t" "h" "t" "h" "h" "t" "h" "t" "t" "h" "h" "h" "h" "h" "h"
[2143] "h" "h" "h" "h" "t" "t" "t" "t" "t" "t" "h" "t" "h" "t" "h" "t" "h" "t"
[2161] "h" "t" "h" "t" "h" "t" "h" "h" "t" "h" "h" "t" "h" "h" "t" "h" "h" "t"
[2179] "h" "h" "t" "h" "h" "t" "h" "h" "t" "h" "h" "h" "h" "h" "h" "h" "h" "h"
[2197] "h" "t" "t" "t" "t" "t" "t" "t" "t" "t" "t" "t" "t" "t" "t" "t" "t" "t"
[2215] "t" "t" "t" "t" "t" "t" "t" "t" "h" "h" "h" "h" "h" "t" "t" "h" "t" "h"
[2233] "t" "h" "t" "h" "h" "t" "h" "t" "t" "t" "t" "t" "t" "t" "t" "t" "t" "t"
[2251] "t" "t" "t" "t" "t" "t" "t" "t" "t" "t" "t" "h" "h" "t" "t" "h" "t" "h"
[2269] "h" "h" "h" "h" "h" "h" "h" "h" "h" "t" "t" "t" "t" "h" "t" "h" "t" "h"
[2287] "t" "h" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "h" "h" "t" "h" "t"
[2305] "h" "t" "h" "h" "t" "h" "t" "h" "h" "t" "h" "t" "h" "t" "h" "h" "h" "h"
[2323] "t" "h" "h" "t" "h" "t" "h" "t" "h" "t" "h" "h" "t" "h" "h" "t" "h" "t"
[2341] "h" "h" "t" "h" "t" "h" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "h"
[2359] "t" "h" "t" "t" "t" "t" "t" "t" "t" "t" "t" "h" "t" "h" "t" "h" "t" "h"
[2377] "t" "h" "t" "h" "h" "t" "h" "t" "h" "h" "t" "h" "t" "t" "h" "t" "h" "t"
[2395] "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "t" "t" "t" "t" "h"
[2413] "t" "h" "t" "h" "t" "t" "h" "t" "h" "t" "h" "h" "h" "t" "t" "h" "h" "h"
[2431] "t" "t" "h" "h" "t" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t"
[2449] "h" "t" "t" "t" "t" "t" "h" "t" "h" "t" "t" "t" "t" "t" "t" "h" "h" "h"
[2467] "h" "t" "t" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "t" "h" "h" "t"
[2485] "h" "t" "h" "t" "h" "t" "t" "h" "t" "h" "t" "h" "t" "t" "t" "h" "h" "t"
[2503] "h" "h" "h" "t" "h" "h" "t" "h" "h" "t" "h" "t" "t" "h" "t" "h" "t" "h"
[2521] "t" "h" "h" "t" "h" "t" "h" "t" "h" "t" "t" "h" "h" "t" "h" "t" "h" "t"
[2539] "h" "t" "h" "t" "t" "t" "h" "h" "h" "t" "h" "h" "h" "t" "h" "t" "t" "t"
[2557] "h" "t" "h" "h" "h" "t" "h" "t" "h" "h" "h" "h" "t" "h" "t" "h" "t" "t"
[2575] "t" "t" "h" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "t" "t" "t"
[2593] "h" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "h" "t" "t" "h" "t" "h"
[2611] "t" "h" "h" "h" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "t" "t" "h" "t"
[2629] "h" "t" "t" "h" "h" "t" "h" "t" "t" "h" "t" "h" "t" "h" "h" "h" "t" "h"
[2647] "t" "h" "h" "h" "t" "h" "t" "h" "h" "h" "t" "t" "t" "t" "h" "h" "h" "t"
[2665] "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "h" "h" "h" "t" "h" "t" "h"
[2683] "h" "t" "h" "h" "h" "h" "h" "h" "t" "h" "t" "h" "h" "t" "h" "h" "h" "h"
[2701] "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "h" "t" "h" "t" "h" "t" "h" "t"
[2719] "h" "h" "t" "t" "t" "t" "t" "h" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t"
[2737] "h" "t" "h" "t" "t" "t" "t" "t" "t" "t" "t" "t" "h" "t" "h" "t" "h" "t"
[2755] "h" "t" "h" "h" "t" "t" "t" "t" "t" "t" "h" "t" "h" "t" "h" "h" "h" "t"
[2773] "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "h" "h" "t" "h" "t" "h" "t"
[2791] "h" "t" "h" "t" "h" "h" "t" "h" "t" "h" "t" "h" "h" "t" "h" "t" "h" "t"
[2809] "h" "t" "h" "t" "h" "t" "h" "h" "t" "h" "t" "h" "t" "h" "t" "h" "h" "h"
[2827] "h" "h" "h" "h" "h" "h" "h" "h" "t" "h" "h" "h" "t" "h" "t" "h" "t" "h"
[2845] "t" "h" "t" "h" "t" "h" "h" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h"
[2863] "h" "t" "h" "t" "h" "t" "h" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h"
[2881] "t" "h" "h" "t" "h" "t" "h" "t" "h" "t" "h" "h" "h" "h" "h" "h" "h" "h"
[2899] "h" "h" "h" "t" "h" "t" "h" "t" "h" "t" "h" "h" "h" "t" "h" "t" "h" "t"
[2917] "t" "h" "t" "h" "t" "h" "t" "t" "h" "t" "h" "t" "t" "t" "h" "t" "t" "h"
[2935] "h" "h" "t" "h" "t" "h" "t" "h" "h" "t" "h" "t" "h" "t" "h" "h" "h" "t"
[2953] "h" "h" "t" "h" "t" "h" "h" "t" "h" "h" "h" "h" "t" "h" "t" "h" "t" "h"
[2971] "h" "h" "t" "t" "t" "t" "t" "t" "t" "h" "h" "h" "t" "h" "t" "h" "t" "h"
[2989] "t" "h" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "h" "t" "h" "h" "t"
[3007] "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t" "t" "t" "t" "t" "t"
[3025] "t" "h" "t" "h" "h" "h" "h" "t" "h" "t" "h" "t" "h" "t" "h" "h" "h" "h"
[3043] "h" "h" "t" "h" "h" "t" "h" "t" "t" "t" "h" "t" "h" "t" "h" "t" "h" "h"
[3061] "t" "h" "t" "t" "t" "t" "t" "t" "t" "t" "t" "t" "h" "t" "h" "h" "h" "h"
[3079] "h" "h" "h" "h" "h" "h" "h" "t" "t" "t" "t" "t" "t" "h" "h" "h" "h" "h"
[3097] "h" "h" "t" "h" "t" "h" "h" "t" "h" "t" "h" "h" "h" "h" "h" "h" "h" "h"
[3115] "h" "h" "t" "h" "t" "h" "h" "h" "h" "h" "h" "t" "h" "t" "h" "t" "h" "h"
[3133] "h" "h" "h" "h" "h" "h" "h" "h" "t" "h" "t" "h" "t" "h" "t" "h" "h" "h"
[3151] "h" "h" "h" "t" "h" "h" "h" "t" "h" "t" "h" "t" "h" "h" "t" "h" "t" "h"
[3169] "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "t" "t"
[3187] "t" "h" "t" "h" "h" "t" "h" "h" "h" "t" "h" "h" "t" "h" "t" "h" "h" "t"
[3205] "h" "h" "t" "h" "h" "h" "h" "h" "h" "t" "h" "h" "t" "h" "t" "h" "h" "h"
[3223] "h" "h" "h" "t" "h" "t" "h" "h" "h" "t" "h" "t" "h" "h" "h" "h" "h" "h"
[3241] "h" "h" "h" "h" "h" "h" "h" "h" "t" "h" "h" "h" "h" "h" "h" "h" "t" "h"
[3259] "h" "h" "h" "t" "h" "t" "h" "t" "h" "h" "h" "h" "h" "h" "t" "t" "t" "t"
[3277] "t" "t" "t" "t" "t" "h" "t" "h" "t" "h" "h" "h" "h" "h" "h" "h" "h" "h"
[3295] "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h"
[3313] "h" "h" "t" "h" "h" "h" "h" "h" "h" "h" "h" "h" "h" "t" "t" "h" "t" "h"
[3331] "t" "h" "t" "h" "t" "h" "h" "t" "h" "t" "t" "t" "t" "h" "h" "t" "h" "t"
[3349] "h" "t" "h" "h" "h" "t" "h" "t" "h" "t" "h" "t" "t" "t" "h" "t" "h" "t"
[3367] "h" "t" "h" "t" "t" "h" "t" "h" "h" "t" "h" "h" "h" "h" "h" "h" "t" "h"
[3385] "t" "h" "t" "h" "h" "h" "h" "h" "h" "t" "h" "t" "h" "t" "h" "t" "h" "t"
[3403] "h" "h" "h" "h" "t" "h" "t" "t" "t" "h" "t" "h" "t" "t" "t" "h" "h" "h"
[3421] "t" "t" "h" "t" "t" "h" "h" "t" "h" "h" "h" "h" "t" "h" "t" "h" "t" "t"
[3439] "h" "h" "h" "t" "t" "t" "t" "h" "h" "h" "h" "h" "h" "t" "h" "t" "h" "t"
[3457] "h" "t" "h" "t" "h" "h" "t" "h" "h" "t" "h" "h" "t" "t" "t" "h" "h" "t"
[3475] "h"How long is this string?
# How many are there?
n <- length(human_flips)Now we need to figure out the length of the runs. The rle() function will make this easy for us, but I’m going to extract the information I need with a custom function called rle_table().
rle_table <- function(x) {
x |>
rle() |>
pluck("lengths") |>
table() |>
as.data.frame() |>
rename(length = Var1) |>
mutate(length = as.numeric(length))
}
# Find the runs
human_runs <- human_flips |>
rle_table()Let’s examine a histogram of the runs
ggplot(data = human_runs, aes(x = length, y = Freq)) +
geom_col()
Pseudorandom flips
Now that we’ve collected the “random” string of T’s and H’s, we’ll use R to create a pseudorandom string of T’s and H’s on the same length. Note the use of the sample() function.
pseudo_flips <- sample(c("t", "h"), size = n, replace = TRUE)
pseudo_runs <- pseudo_flips |>
rle_table()Probability theory
Finally, we’ll add a third method for computing streak lengths. The following data frame comes directly from probability theory.
# Expected number of runs of length k
random_runs <- human_runs |>
full_join(pseudo_runs, by = join_by(length)) |>
arrange(length)
random_runs <- random_runs |>
mutate(pr_k = (n - length - 1) * 2 * (1/2)^(length+2))
random_runs length Freq.x Freq.y pr_k
1 1 1645 822 8.682500e+02
2 2 275 455 4.340000e+02
3 3 91 229 2.169375e+02
4 4 39 98 1.084375e+02
5 5 21 55 5.420312e+01
6 6 24 25 2.709375e+01
7 7 10 16 1.354297e+01
8 8 8 11 6.769531e+00
9 9 11 2 3.383789e+00
10 10 10 1 1.691406e+00
11 11 4 1 8.454590e-01
12 12 2 NA 4.226074e-01
13 13 1 NA 2.112427e-01
14 14 2 NA 1.055908e-01
15 15 1 NA 5.278015e-02
16 16 2 NA 2.638245e-02
17 17 1 NA 1.318741e-02
18 18 1 NA 6.591797e-03
19 19 1 NA 3.294945e-03
20 20 1 NA 1.646996e-03
21 21 1 NA 8.232594e-04Finally, plot all three frequencies. Which one is which?????
# Plot the frequency of runs for both samples
random_runs |>
pivot_longer(-length, names_to = "src", values_to = "freq") |>
ggplot(aes(x = length, y = freq, color = src)) +
geom_line() +
scale_x_continuous("Length of Run") +
scale_y_continuous("Frequency of Run") +
scale_color_discrete(guide = "none") +
ggtitle(paste("Number of Flips =", n))