I think that James Holzhauer lost on purpose against Emma, in his last game. My evidence is that he had been wagering big every game up until then and suddenly he wagers low against Emma. I suspect the producers want Ken Jennings to host the show after Alex steps down. The show would be more dramatic if the host had the records in both shows and money won. Thus, they paid off James to throw the game.

Tanko

Let me set the stage. On the June 3, 2019 James was within easy reach of breaking the record for total money won in regular games, which still stands at \$2,520,700. James average win per game was much more than what he needed to break the record. So all eyes were watching on June 3 to see the record broken.

Instead, what happens is not only does James not break the record, but he loses. The winner, Emma, played a very strong strategic game as well as being good with the buzzer and simply answering correctly. She played just as James normally did. Going into Final Jeopardy the scores were:

• Emma — \$26,600
• James — \$23,400
• Jay — \$11,000

In these situations, where second place has more than half of first place, and third place does not, it typically comes down to first and second place choosing to go high or low with their final wager. A high wager for first place is enough to lock in a win if correct. To be specific, two times the second place score less the first place score plus one dollar. That is exactly what Emma did with a wager of 2×\$23,400 - \$26,600 + \$1 = \$20,201. Most of the time, the first place player does this.

However, James didn't know what Emma would do when deciding his wager. The following table shows who would win according to what combination of wagers.

Click on image for larger version.

If Emma wagers at least \$20,201 then she locks in a win if correct.

If Emma wagers low then she will win if either (a) James wagers low or (b) James wagers high and is wrong.

If James wagers high then he wins if (a) Emma goes high, Emma is wrong, and James is right, or (b) Emma goes low and James is right.

If James wagers low then he wins if Emma goes high and is wrong.

If perfect logicians were playing, both would randomize their decisions. However, rarely does the leader go low in these situations where he/she can be caught. If James anticipates Emma to go high, then he absolutely should go low. This way he doesn't have to get Final Jeopardy right to win, he just has to hope Emma blows it.

James actual bid was the correct amount to cover Jay if Jay bet everything and was right: \$23,400 - 2×\$11,000 - \$1 = \$1,399, which satisfied as a low wager for purposes of beating Emma.

If correct, James would get an extra \$1,000 for coming in second, compared to third.

In conclusion, I completely reject the conspiracy theory that James threw the game. He played the right way and lost due to a combination of playing a strong competitor and what most people would call "bad luck."

In honor of Ask the Wizard column #314, what are your favorite infinite series that sum to some function of pi?

Heather

It is easy to choose these two, as probably the two most famous:

• 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ... = π/4
• 1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 + ... = π^2/6

I found a blackjack machine with a player advantage of 0.1%. As a group, we've played flat bet 2,015,000 hands and am down 1,475 units. I don't specify the exact rules, lest I give the play away, so please take the 0.1% player advantage on faith. What are the odds of running as bad as we are, assuming the game is fair?

anonymous

Based on that advantage and hands played, the expected win is 2015 units. Assuming a standard deviation of 1.1 per hand, a standard deviation on the entire play is 1,561. The difference between your actual win and expected win is 3,490. That is 3,490/1,561 = 2.24 standard deviations. The probability of results that bad or worse is 1.27%.

I've heard that with 23 random people, the probability of at least one common birthday to two or more people is over 50%? This this true? What is the probability of a common birthday for other group sizes? The same question also for a common birthday to 3, 4, and 5 people.

anonymous

That is true, with 23 random people, the probability of at least one pair of people have a common birthday is 50.73%. This ignores leap day and assumes everybody has an equal chance of being born on each of the other 365 days (which is not actually the case, spring and fall birthdays are slightly more common).

The tables in answer to your question are quote long, so I'll put them in spoiler tags. Click on the buttons for the answers.

### Common Birthday to 2+ People

Group Size Probability
2 0.002740
3 0.008204
4 0.016356
5 0.027136
6 0.040462
7 0.056236
8 0.074335
9 0.094624
10 0.116948
11 0.141141
12 0.167025
13 0.194410
14 0.223103
15 0.252901
16 0.283604
17 0.315008
18 0.346911
19 0.379119
20 0.411438
21 0.443688
22 0.475695
23 0.507297
24 0.538344
25 0.568700
26 0.598241
27 0.626859
28 0.654461
29 0.680969
30 0.706316
31 0.730455
32 0.753348
33 0.774972
34 0.795317
35 0.814383
36 0.832182
37 0.848734
38 0.864068
39 0.878220
40 0.891232
41 0.903152
42 0.914030
43 0.923923
44 0.932885
45 0.940976
46 0.948253
47 0.954774
48 0.960598
49 0.965780
50 0.970374
51 0.974432
52 0.978005
53 0.981138
54 0.983877
55 0.986262
56 0.988332
57 0.990122
58 0.991665
59 0.992989
60 0.994123
61 0.995089
62 0.995910
63 0.996604
64 0.997190
65 0.997683
66 0.998096
67 0.998440
68 0.998726
69 0.998964
70 0.999160
71 0.999321
72 0.999453
73 0.999561
74 0.999649
75 0.999720
76 0.999777
77 0.999824
78 0.999861
79 0.999891
80 0.999914
81 0.999933
82 0.999948
83 0.999960
84 0.999969

### Common Birthday to 3+ People

Group Size Probability
3 0.000008
4 0.000030
5 0.000075
6 0.000149
7 0.000261
8 0.000416
9 0.000623
10 0.000888
11 0.001218
12 0.001621
13 0.002102
14 0.002670
15 0.003329
16 0.004088
17 0.004953
18 0.005929
19 0.007024
20 0.008243
21 0.009592
22 0.011078
23 0.012705
24 0.014481
25 0.016409
26 0.018497
27 0.020747
28 0.023167
29 0.025760
30 0.028531
31 0.031484
32 0.034624
33 0.037954
34 0.041479
35 0.045202
36 0.049126
37 0.053254
38 0.057589
39 0.062133
40 0.066889
41 0.071859
42 0.077044
43 0.082446
44 0.088065
45 0.093903
46 0.099960
47 0.106236
48 0.112731
49 0.119444
50 0.126375
51 0.133522
52 0.140885
53 0.148460
54 0.156246
55 0.164241
56 0.172441
57 0.180844
58 0.189445
59 0.198242
60 0.207230
61 0.216405
62 0.225761
63 0.235294
64 0.244999
65 0.254869
66 0.264899
67 0.275082
68 0.285413
69 0.295883
70 0.306487
71 0.317217
72 0.328066
73 0.339026
74 0.350088
75 0.361246
76 0.372491
77 0.383814
78 0.395207
79 0.406662
80 0.418169
81 0.429720
82 0.441307
83 0.452920
84 0.464550
85 0.476188
86 0.487826
87 0.499455
88 0.511065
89 0.522648
90 0.534196
91 0.545698
92 0.557148
93 0.568537
94 0.579855
95 0.591096
96 0.602252
97 0.613314
98 0.624275
99 0.635127
100 0.645865
101 0.656480
102 0.666967
103 0.677318
104 0.687529
105 0.697593
106 0.707505
107 0.717260
108 0.726853
109 0.736279
110 0.745536
111 0.754619
112 0.763525
113 0.772251
114 0.780795
115 0.789155
116 0.797330
117 0.805319
118 0.813121
119 0.820580
120 0.827964
121 0.835152
122 0.842144
123 0.848940
124 0.855540
125 0.861945
126 0.868155
127 0.874172
128 0.879996
129 0.885631
130 0.891076
131 0.896335
132 0.901409
133 0.906302
134 0.911015
135 0.915552
136 0.919915
137 0.924108
138 0.928135
139 0.931997
140 0.935700
141 0.939246
142 0.942640
143 0.945885
144 0.948985
145 0.951944
146 0.954766
147 0.957456
148 0.960016
149 0.962452
150 0.964767
151 0.966965
152 0.969050
153 0.971028
154 0.972900
155 0.974672
156 0.976347
157 0.977930
158 0.979423
159 0.980831
160 0.982158
161 0.983407
162 0.984581
163 0.985684
164 0.986719
165 0.987690
166 0.988600
167 0.989452
168 0.990248
169 0.990992
170 0.991687
171 0.992335
172 0.992938
173 0.993500
174 0.994022
175 0.994508
176 0.994958
177 0.995376
178 0.995763
179 0.996121
180 0.996452
181 0.996758
182 0.997040
183 0.997300
184 0.997540
185 0.997760
186 0.997963
187 0.998149
188 0.998319
189 0.998476
190 0.998619
191 0.998750
192 0.998869
193 0.998979
194 0.999078
195 0.999169
196 0.999251
197 0.999326
198 0.999394
199 0.999456
200 0.999512
201 0.999562
202 0.999608
203 0.999650
204 0.999687
205 0.999720
206 0.999751
207 0.999778
208 0.999802
209 0.999824
210 0.999844
211 0.999862
212 0.999877
213 0.999891
214 0.999904
215 0.999915
216 0.999925
217 0.999934
218 0.999942
219 0.999949
220 0.999955
221 0.999961
222 0.999966
223 0.999970
224 0.999974
225 0.999977
226 0.999980
227 0.999982
228 0.999985
229 0.999987
230 0.999988
231 0.999990
232 0.999991
233 0.999992
234 0.999994
235 0.999994
236 0.999995
237 0.999996
238 0.999996
239 0.999997
240 0.999997
241 0.999998
242 0.999998
243 0.999998
244 0.999999

### Common Birthday to 4+ People

Group Size Probability
4 0.000000
5 0.000000
6 0.000000
7 0.000001
8 0.000001
9 0.000003
10 0.000004
11 0.000007
12 0.000010
13 0.000014
14 0.000020
15 0.000027
16 0.000036
17 0.000048
18 0.000061
19 0.000077
20 0.000096
21 0.000119
22 0.000145
23 0.000175
24 0.000209
25 0.000248
26 0.000293
27 0.000343
28 0.000399
29 0.000462
30 0.000532
31 0.000610
32 0.000695
33 0.000790
34 0.000893
35 0.001006
36 0.001129
37 0.001263
38 0.001408
39 0.001566
40 0.001736
41 0.001919
42 0.002116
43 0.002328
44 0.002555
45 0.002798
46 0.003058
47 0.003334
48 0.003629
49 0.003943
50 0.004276
51 0.004629
52 0.005003
53 0.005399
54 0.005817
55 0.006258
56 0.006724
57 0.007214
58 0.007730
59 0.008272
60 0.008841
61 0.009439
62 0.010065
63 0.010721
64 0.011408
65 0.012126
66 0.012876
67 0.013659
68 0.014476
69 0.015327
70 0.016215
71 0.017139
72 0.018100
73 0.019099
74 0.020137
75 0.021215
76 0.022334
77 0.023495
78 0.024698
79 0.025944
80 0.027235
81 0.028570
82 0.029951
83 0.031379
84 0.032855
85 0.034379
86 0.035952
87 0.037575
88 0.039249
89 0.040974
90 0.042752
91 0.044583
92 0.046467
93 0.048407
94 0.050402
95 0.052453
96 0.054561
97 0.056726
98 0.058950
99 0.061233
100 0.063576
101 0.065978
102 0.068442
103 0.070967
104 0.073554
105 0.076204
106 0.078917
107 0.081694
108 0.084535
109 0.087441
110 0.090412
111 0.093449
112 0.096552
113 0.099722
114 0.102958
115 0.106262
116 0.109633
117 0.113072
118 0.116579
119 0.120154
120 0.123798
121 0.127510
122 0.131292
123 0.135142
124 0.139061
125 0.143050
126 0.147107
127 0.151234
128 0.155429
129 0.159694
130 0.164027
131 0.168429
132 0.172899
133 0.177438
134 0.182044
135 0.186719
136 0.191460
137 0.196269
138 0.201144
139 0.206085
140 0.211091
141 0.216163
142 0.221299
143 0.226499
144 0.231763
145 0.237089
146 0.242476
147 0.247925
148 0.253434
149 0.259002
150 0.264629
151 0.270314
152 0.276055
153 0.281852
154 0.287703
155 0.293608
156 0.299566
157 0.305575
158 0.311634
159 0.317741
160 0.323897
161 0.330099
162 0.336346
163 0.342637
164 0.348970
165 0.355343
166 0.361757
167 0.368208
168 0.374696
169 0.381218
170 0.387774
171 0.394362
172 0.400980
173 0.407626
174 0.414299
175 0.420997
176 0.427718
177 0.434462
178 0.441224
179 0.448005
180 0.454803
181 0.461615
182 0.468439
183 0.475274
184 0.482118
185 0.488969
186 0.495826
187 0.502685
188 0.509546
189 0.516407
190 0.523265
191 0.530119
192 0.536967
193 0.543807
194 0.550636
195 0.557454
196 0.564258
197 0.571046
198 0.577817
199 0.584568
200 0.591298
201 0.598005
202 0.604687
203 0.611342
204 0.617969
205 0.624565
206 0.631129
207 0.637659
208 0.644154
209 0.650611
210 0.657030
211 0.663407
212 0.669743
213 0.676035
214 0.682281
215 0.688481
216 0.694632
217 0.700734
218 0.706784
219 0.712782
220 0.718726
221 0.724614
222 0.730446
223 0.736220
224 0.741936
225 0.747591
226 0.753185
227 0.758717
228 0.764185
229 0.769590
230 0.774929
231 0.780202
232 0.785409
233 0.790547
234 0.795618
235 0.800619
236 0.805551
237 0.810412
238 0.815202
239 0.819921
240 0.824569
241 0.829144
242 0.833646
243 0.838076
244 0.842432
245 0.846716
246 0.850925
247 0.855061
248 0.859123
249 0.863112
250 0.867027
251 0.870868
252 0.874635
253 0.878329
254 0.881950
255 0.885498
256 0.888973
257 0.892375
258 0.895705
259 0.898964
260 0.902151
261 0.905268
262 0.908314
263 0.911290
264 0.914197
265 0.917036
266 0.919806
267 0.922509
268 0.925145
269 0.927715
270 0.930220
271 0.932661
272 0.935037
273 0.937351
274 0.939603
275 0.941793
276 0.943923
277 0.945993
278 0.948005
279 0.949960
280 0.951857
281 0.953699
282 0.955486
283 0.957218
284 0.958898
285 0.960527
286 0.962104
287 0.963631
288 0.965109
289 0.966540
290 0.967923
291 0.969260
292 0.970553
293 0.971802
294 0.973007
295 0.974171
296 0.975294
297 0.976377
298 0.977421
299 0.978427
300 0.979397
301 0.980330
302 0.981228
303 0.982092
304 0.982923
305 0.983722
306 0.984490
307 0.985227
308 0.985935
309 0.986614
310 0.987266
311 0.987890
312 0.988489
313 0.989063
314 0.989612
315 0.990138
316 0.990641
317 0.991122
318 0.991581
319 0.992021
320 0.992440
321 0.992841
322 0.993223
323 0.993587
324 0.993935
325 0.994266
326 0.994581
327 0.994882
328 0.995167
329 0.995439
330 0.995698
331 0.995943
332 0.996176
333 0.996398
334 0.996608
335 0.996807
336 0.996996
337 0.997175
338 0.997344
339 0.997505
340 0.997657
341 0.997801
342 0.997936
343 0.998065
344 0.998186
345 0.998300
346 0.998408
347 0.998510
348 0.998606
349 0.998696
350 0.998781
351 0.998861
352 0.998937
353 0.999008
354 0.999074
355 0.999137
356 0.999195
357 0.999250
358 0.999302
359 0.999350
360 0.999396
361 0.999438
362 0.999478
363 0.999515
364 0.999550
365 0.999582
366 0.999613
367 0.999641
368 0.999668
369 0.999692
370 0.999715
371 0.999736
372 0.999756
373 0.999775
374 0.999792
375 0.999808
376 0.999823
377 0.999837
378 0.999850
379 0.999861
380 0.999872
381 0.999883
382 0.999892
383 0.999901
384 0.999909
385 0.999916
386 0.999923
387 0.999930
388 0.999935
389 0.999941
390 0.999946
391 0.999950
392 0.999955
393 0.999959
394 0.999962
395 0.999965
396 0.999969
397 0.999971
398 0.999974
399 0.999976
400 0.999978
401 0.999980
402 0.999982
403 0.999984
404 0.999985
405 0.999987
406 0.999988
407 0.999989
408 0.999990
409 0.999991
410 0.999992
411 0.999993
412 0.999993
413 0.999994
414 0.999995
415 0.999995
416 0.999996
417 0.999996
418 0.999996
419 0.999997
420 0.999997
421 0.999997
422 0.999998
423 0.999998
424 0.999998
425 0.999998
426 0.999998
427 0.999999
428 0.999999
429 0.999999

### Common Birthday to 5+ People

Group Size Probability
5 0.000000
6 0.000000
7 0.000000
8 0.000000
9 0.000000
10 0.000000
11 0.000000
12 0.000000
13 0.000000
14 0.000000
15 0.000000
16 0.000000
17 0.000000
18 0.000001
19 0.000001
20 0.000001
21 0.000001
22 0.000002
23 0.000002
24 0.000003
25 0.000004
26 0.000004
27 0.000005
28 0.000006
29 0.000008
30 0.000009
31 0.000011
32 0.000013
33 0.000015
34 0.000017
35 0.000020
36 0.000023
37 0.000026
38 0.000030
39 0.000034
40 0.000039
41 0.000044
42 0.000050
43 0.000056
44 0.000063
45 0.000070
46 0.000079
47 0.000087
48 0.000097
49 0.000108
50 0.000119
51 0.000132
52 0.000145
53 0.000159
54 0.000175
55 0.000192
56 0.000209
57 0.000229
58 0.000249
59 0.000271
60 0.000295
61 0.000320
62 0.000347
63 0.000375
64 0.000406
65 0.000438
66 0.000472
67 0.000509
68 0.000547
69 0.000588
70 0.000631
71 0.000676
72 0.000725
73 0.000775
74 0.000829
75 0.000885
76 0.000944
77 0.001007
78 0.001072
79 0.001141
80 0.001213
81 0.001289
82 0.001369
83 0.001452
84 0.001539
85 0.001630
86 0.001726
87 0.001825
88 0.001930
89 0.002038
90 0.002152
91 0.002270
92 0.002394
93 0.002522
94 0.002656
95 0.002796
96 0.002941
97 0.003092
98 0.003249
99 0.003412
100 0.003581
101 0.003757
102 0.003939
103 0.004128
104 0.004325
105 0.004528
106 0.004739
107 0.004957
108 0.005183
109 0.005417
110 0.005659
111 0.005909
112 0.006168
113 0.006436
114 0.006712
115 0.006998
116 0.007293
117 0.007597
118 0.007912
119 0.008236
120 0.008570
121 0.008915
122 0.009270
123 0.009636
124 0.010013
125 0.010402
126 0.010801
127 0.011213
128 0.011637
129 0.012072
130 0.012521
131 0.012981
132 0.013455
133 0.013942
134 0.014442
135 0.014956
136 0.015484
137 0.016026
138 0.016582
139 0.017153
140 0.017739
141 0.018340
142 0.018956
143 0.019588
144 0.020235
145 0.020899
146 0.021580
147 0.022277
148 0.022991
149 0.023722
150 0.024470
151 0.025237
152 0.026021
153 0.026824
154 0.027645
155 0.028485
156 0.029344
157 0.030222
158 0.031120
159 0.032037
160 0.032975
161 0.033934
162 0.034913
163 0.035912
164 0.036934
165 0.037976
166 0.039040
167 0.040127
168 0.041235
169 0.042367
170 0.043521
171 0.044698
172 0.045898
173 0.047122
174 0.048370
175 0.049642
176 0.050939
177 0.052260
178 0.053606
179 0.054977
180 0.056374
181 0.057796
182 0.059245
183 0.060719
184 0.062220
185 0.063748
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