Statistics on 165 successive \pdfuniformdeviate 2. It seems they is `>50%` chance of more 1s than 0s, independently of starting point.

RNGwalkhighbit.out.txt

HIGH BIT (\pdfuniformdeviate 2)

In successive batches of 165 draws, are there are more 1s than 0s
The answer is Yes on average for all initial shifts and the two
randomly chosen seeds 149847074 and 256844633

(I should have taken the first one < 2**27, too late)


+++++++++++++++++++++++
10000 runs of 165 draws
+++++++++++++++++++++++

(shift=1, seed=149847074) 0:4889, 1:5111
(shift=1, seed=256844633) 0:4895, 1:5105

(shift=2, seed=149847074) 0:4893, 1:5107
(shift=2, seed=256844633) 0:4903, 1:5097

(shift=3, seed=149847074) 0:4900, 1:5100
(shift=3, seed=256844633) 0:4903, 1:5097

(shift=4, seed=149847074) 0:4890, 1:5110
(shift=4, seed=256844633) 0:4930, 1:5070

(shift=5, seed=149847074) 0:4906, 1:5094
(shift=5, seed=256844633) 0:4930, 1:5070

(shift=6, seed=149847074) 0:4922, 1:5078
(shift=6, seed=256844633) 0:4899, 1:5101

(shift=7, seed=149847074) 0:4913, 1:5087
(shift=7, seed=256844633) 0:4908, 1:5092

(shift=8, seed=149847074) 0:4920, 1:5080
(shift=8, seed=256844633) 0:4917, 1:5083

(shift=9, seed=149847074) 0:4944, 1:5056
(shift=9, seed=256844633) 0:4916, 1:5084

(shift=10, seed=149847074) 0:4932, 1:5068
(shift=10, seed=256844633) 0:4932, 1:5068

(shift=11, seed=149847074) 0:4905, 1:5095
(shift=11, seed=256844633) 0:4923, 1:5077

(shift=12, seed=149847074) 0:4928, 1:5072
(shift=12, seed=256844633) 0:4937, 1:5063

(shift=13, seed=149847074) 0:4934, 1:5066
(shift=13, seed=256844633) 0:4938, 1:5062

(shift=14, seed=149847074) 0:4900, 1:5100
(shift=14, seed=256844633) 0:4940, 1:5060

(shift=15, seed=149847074) 0:4911, 1:5089
(shift=15, seed=256844633) 0:4949, 1:5051

(shift=16, seed=149847074) 0:4928, 1:5072
(shift=16, seed=256844633) 0:4939, 1:5061

(shift=17, seed=149847074) 0:4932, 1:5068
(shift=17, seed=256844633) 0:4949, 1:5051

(shift=18, seed=149847074) 0:4926, 1:5074
(shift=18, seed=256844633) 0:4944, 1:5056

(shift=19, seed=149847074) 0:4926, 1:5074
(shift=19, seed=256844633) 0:4951, 1:5049

(shift=20, seed=149847074) 0:4918, 1:5082
(shift=20, seed=256844633) 0:4949, 1:5051

(shift=21, seed=149847074) 0:4937, 1:5063
(shift=21, seed=256844633) 0:4938, 1:5062

(shift=22, seed=149847074) 0:4941, 1:5059
(shift=22, seed=256844633) 0:4919, 1:5081

(shift=23, seed=149847074) 0:4955, 1:5045
(shift=23, seed=256844633) 0:4912, 1:5088

(shift=24, seed=149847074) 0:4935, 1:5065
(shift=24, seed=256844633) 0:4923, 1:5077

(shift=25, seed=149847074) 0:4952, 1:5048
(shift=25, seed=256844633) 0:4916, 1:5084

(shift=26, seed=149847074) 0:4964, 1:5036
(shift=26, seed=256844633) 0:4928, 1:5072

(shift=27, seed=149847074) 0:4952, 1:5048
(shift=27, seed=256844633) 0:4970, 1:5030

(shift=28, seed=149847074) 0:4976, 1:5024
(shift=28, seed=256844633) 0:4942, 1:5058

(shift=29, seed=149847074) 0:4944, 1:5056
(shift=29, seed=256844633) 0:4923, 1:5077

(shift=30, seed=149847074) 0:4970, 1:5030
(shift=30, seed=256844633) 0:4937, 1:5063

(shift=31, seed=149847074) 0:4944, 1:5056
(shift=31, seed=256844633) 0:4945, 1:5055

(shift=32, seed=149847074) 0:4971, 1:5029
(shift=32, seed=256844633) 0:4937, 1:5063

(shift=33, seed=149847074) 0:4954, 1:5046
(shift=33, seed=256844633) 0:4951, 1:5049

(shift=34, seed=149847074) 0:4938, 1:5062
(shift=34, seed=256844633) 0:4959, 1:5041

(shift=35, seed=149847074) 0:4938, 1:5062
(shift=35, seed=256844633) 0:4924, 1:5076

(shift=36, seed=149847074) 0:4949, 1:5051
(shift=36, seed=256844633) 0:4913, 1:5087

(shift=37, seed=149847074) 0:4896, 1:5104
(shift=37, seed=256844633) 0:4946, 1:5054

(shift=38, seed=149847074) 0:4886, 1:5114
(shift=38, seed=256844633) 0:4941, 1:5059

(shift=39, seed=149847074) 0:4895, 1:5105
(shift=39, seed=256844633) 0:4948, 1:5052

(shift=40, seed=149847074) 0:4889, 1:5111
(shift=40, seed=256844633) 0:4926, 1:5074

(shift=41, seed=149847074) 0:4866, 1:5134
(shift=41, seed=256844633) 0:4916, 1:5084

(shift=42, seed=149847074) 0:4884, 1:5116
(shift=42, seed=256844633) 0:4922, 1:5078

(shift=43, seed=149847074) 0:4886, 1:5114
(shift=43, seed=256844633) 0:4913, 1:5087

(shift=44, seed=149847074) 0:4923, 1:5077
(shift=44, seed=256844633) 0:4891, 1:5109

(shift=45, seed=149847074) 0:4935, 1:5065
(shift=45, seed=256844633) 0:4912, 1:5088

(shift=46, seed=149847074) 0:4924, 1:5076
(shift=46, seed=256844633) 0:4938, 1:5062

(shift=47, seed=149847074) 0:4910, 1:5090
(shift=47, seed=256844633) 0:4950, 1:5050

(shift=48, seed=149847074) 0:4913, 1:5087
(shift=48, seed=256844633) 0:4909, 1:5091

(shift=49, seed=149847074) 0:4923, 1:5077
(shift=49, seed=256844633) 0:4890, 1:5110

(shift=50, seed=149847074) 0:4898, 1:5102
(shift=50, seed=256844633) 0:4857, 1:5143

(shift=51, seed=149847074) 0:4898, 1:5102
(shift=51, seed=256844633) 0:4880, 1:5120

(shift=52, seed=149847074) 0:4874, 1:5126
(shift=52, seed=256844633) 0:4878, 1:5122

(shift=53, seed=149847074) 0:4887, 1:5113
(shift=53, seed=256844633) 0:4871, 1:5129

(shift=54, seed=149847074) 0:4868, 1:5132
(shift=54, seed=256844633) 0:4887, 1:5113

(shift=55, seed=149847074) 0:4874, 1:5126
(shift=55, seed=256844633) 0:4875, 1:5125


++++++++++++++++++++++
30000 runs of 55 draws
++++++++++++++++++++++

(shift=1, seed=149847074) 0:14959, 1:15041
(shift=1, seed=256844633) 0:15025, 1:14975

(shift=2, seed=149847074) 0:14952, 1:15048
(shift=2, seed=256844633) 0:15037, 1:14963

(shift=3, seed=149847074) 0:14940, 1:15060
(shift=3, seed=256844633) 0:14958, 1:15042

(shift=4, seed=149847074) 0:14909, 1:15091
(shift=4, seed=256844633) 0:15038, 1:14962

(shift=5, seed=149847074) 0:14950, 1:15050
(shift=5, seed=256844633) 0:15014, 1:14986

(shift=6, seed=149847074) 0:14963, 1:15037
(shift=6, seed=256844633) 0:15059, 1:14941

(shift=7, seed=149847074) 0:14949, 1:15051
(shift=7, seed=256844633) 0:15039, 1:14961

(shift=8, seed=149847074) 0:14938, 1:15062
(shift=8, seed=256844633) 0:15050, 1:14950

(shift=9, seed=149847074) 0:14917, 1:15083
(shift=9, seed=256844633) 0:15015, 1:14985

(shift=10, seed=149847074) 0:14947, 1:15053
(shift=10, seed=256844633) 0:15074, 1:14926

(shift=11, seed=149847074) 0:15015, 1:14985
(shift=11, seed=256844633) 0:15059, 1:14941

(shift=12, seed=149847074) 0:15006, 1:14994
(shift=12, seed=256844633) 0:15049, 1:14951

(shift=13, seed=149847074) 0:15031, 1:14969
(shift=13, seed=256844633) 0:15034, 1:14966

(shift=14, seed=149847074) 0:15061, 1:14939
(shift=14, seed=256844633) 0:15104, 1:14896

(shift=15, seed=149847074) 0:15083, 1:14917
(shift=15, seed=256844633) 0:15021, 1:14979

(shift=16, seed=149847074) 0:15025, 1:14975
(shift=16, seed=256844633) 0:15040, 1:14960

(shift=17, seed=149847074) 0:15021, 1:14979
(shift=17, seed=256844633) 0:15060, 1:14940

(shift=18, seed=149847074) 0:14963, 1:15037
(shift=18, seed=256844633) 0:15104, 1:14896

(shift=19, seed=149847074) 0:14975, 1:15025
(shift=19, seed=256844633) 0:15102, 1:14898

(shift=20, seed=149847074) 0:15017, 1:14983
(shift=20, seed=256844633) 0:15076, 1:14924

(shift=21, seed=149847074) 0:15028, 1:14972
(shift=21, seed=256844633) 0:15055, 1:14945

(shift=22, seed=149847074) 0:15015, 1:14985
(shift=22, seed=256844633) 0:15019, 1:14981

(shift=23, seed=149847074) 0:15046, 1:14954
(shift=23, seed=256844633) 0:15047, 1:14953

(shift=24, seed=149847074) 0:15002, 1:14998
(shift=24, seed=256844633) 0:14976, 1:15024

(shift=25, seed=149847074) 0:14996, 1:15004
(shift=25, seed=256844633) 0:14995, 1:15005

(shift=26, seed=149847074) 0:14956, 1:15044
(shift=26, seed=256844633) 0:15033, 1:14967

(shift=27, seed=149847074) 0:14904, 1:15096
(shift=27, seed=256844633) 0:14967, 1:15033

(shift=28, seed=149847074) 0:14943, 1:15057
(shift=28, seed=256844633) 0:15016, 1:14984

(shift=29, seed=149847074) 0:14980, 1:15020
(shift=29, seed=256844633) 0:14986, 1:15014

(shift=30, seed=149847074) 0:14991, 1:15009
(shift=30, seed=256844633) 0:14994, 1:15006

(shift=31, seed=149847074) 0:14975, 1:15025
(shift=31, seed=256844633) 0:14979, 1:15021

(shift=32, seed=149847074) 0:15042, 1:14958
(shift=32, seed=256844633) 0:14985, 1:15015

(shift=33, seed=149847074) 0:15019, 1:14981
(shift=33, seed=256844633) 0:14996, 1:15004

(shift=34, seed=149847074) 0:15021, 1:14979
(shift=34, seed=256844633) 0:14970, 1:15030

(shift=35, seed=149847074) 0:15033, 1:14967
(shift=35, seed=256844633) 0:14953, 1:15047

(shift=36, seed=149847074) 0:14999, 1:15001
(shift=36, seed=256844633) 0:14967, 1:15033

(shift=37, seed=149847074) 0:14981, 1:15019
(shift=37, seed=256844633) 0:15000, 1:15000

(shift=38, seed=149847074) 0:14889, 1:15111
(shift=38, seed=256844633) 0:14997, 1:15003

(shift=39, seed=149847074) 0:14905, 1:15095
(shift=39, seed=256844633) 0:15018, 1:14982

(shift=40, seed=149847074) 0:14882, 1:15118
(shift=40, seed=256844633) 0:15063, 1:14937

(shift=41, seed=149847074) 0:14871, 1:15129
(shift=41, seed=256844633) 0:15021, 1:14979

(shift=42, seed=149847074) 0:14877, 1:15123
(shift=42, seed=256844633) 0:15000, 1:15000

(shift=43, seed=149847074) 0:14935, 1:15065
(shift=43, seed=256844633) 0:15007, 1:14993

(shift=44, seed=149847074) 0:14890, 1:15110
(shift=44, seed=256844633) 0:15014, 1:14986

(shift=45, seed=149847074) 0:14920, 1:15080
(shift=45, seed=256844633) 0:15060, 1:14940

(shift=46, seed=149847074) 0:14940, 1:15060
(shift=46, seed=256844633) 0:15057, 1:14943

(shift=47, seed=149847074) 0:14913, 1:15087
(shift=47, seed=256844633) 0:15078, 1:14922

(shift=48, seed=149847074) 0:14905, 1:15095
(shift=48, seed=256844633) 0:15023, 1:14977

(shift=49, seed=149847074) 0:14895, 1:15105
(shift=49, seed=256844633) 0:14993, 1:15007

(shift=50, seed=149847074) 0:14931, 1:15069
(shift=50, seed=256844633) 0:14990, 1:15010

(shift=51, seed=149847074) 0:15010, 1:14990
(shift=51, seed=256844633) 0:15041, 1:14959

(shift=52, seed=149847074) 0:14912, 1:15088
(shift=52, seed=256844633) 0:15015, 1:14985

(shift=53, seed=149847074) 0:14928, 1:15072
(shift=53, seed=256844633) 0:14969, 1:15031

(shift=54, seed=149847074) 0:14946, 1:15054
(shift=54, seed=256844633) 0:14990, 1:15010

(shift=55, seed=149847074) 0:14895, 1:15105
(shift=55, seed=256844633) 0:14964, 1:15036