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Ask the Slot Expert: Son of Monty Hall plays Survivor

25 May 2022

The odds don't work that way.

The odds ARE 50/50 once one of the losers is revealed. The odds change, just like in Blackjack once cards are revealed. Otherwise, counting cards would be meaningless, and we know that's not the case.

I thought of a couple of alternate scenarios, but I won't bore you with the machinations of having more than 3 choices, and the reality is that you had a 1 in 3 chance at the beginning, and that doesn't change, but there was also only a 1 in 3 chance that the particular remaining "non-choice" was the good one. Therefore, your chances are equal.

Answer: You're right that you had a 1 in 3 chance of selecting the good box. You're also right that that doesn't change, but then you argue that it changes to 1 in 2.

Well, actually you said that the "chances are equal" because each box had a 1 in 3 chance of being good in the beginning of the game. If the chosen box has a 1 in 3 chance of being good and the remaining unchosen box has a 1 in 3 chance, where did the other 1 in 3 chance go? Probabilities have to add to 1.

Remember how the game works.

• We are told that there is a prize in one of three boxes.
• We choose one of the boxes.
• The omniscient host reveals one of the boxes that we didn't choose. The box is bad.
• We are then asked if we want to stick with the box we chose or switch to the other box.

When we choose a box, we partition the boxes into two groups. Each box has a 1 in 3 chance of having the prize. The chosen group has a 1 in 3 chance of being good. The unchosen group has two boxes, so that group has a 2 in 3 chance of having the good box in it. These chances don't change.

Let's play Einstein and do some thought experiments. First, let's choose one box at random and see what's inside. Note the result. Close the box. Mix them up. Choose again. Repeat.

Won't we have chosen the good box one-third of the time? How does revealing what's inside one of the boxes we didn't choose affect that?

Let's play again, but remove the histrionics of opening one of the unchosen boxes. Choose a box and then open all of them at once.

Won't we have chosen the good box one-third of the time? And won't the unchosen group have the good box two-thirds of the time?

The following table shows the three possibilities. After the host reveals that one of the boxes in the unchosen group is bad, which row can we eliminate?

ChosenUnchosenStickSwitch
GBBWinLose
BGBLoseWin
BBGLoseWin

Einstein had to rely on thought experiments because he couldn't travel near the speed of light, but we don't have to rely on them. We can play the game ourselves.

Try it. I did.

I took three slips of paper. I wrote G on one of them and B on the other two. I "shuffled" the slips under a table so I couldn't see them and picked two of them at random. I then looked at what was on them. I hope you'll agree that not revealing what is on one of the slips has no effect on what is on the other slip.

Picking two slips is equivalent to switching to the unchosen group of two boxes in the game. I did this 100 times.

Here are my results: I won 65 times and lost 35.

The good box with the prize is in the unchosen group of two boxes two-thirds of time and the good box is in the chosen group of one box one-third of the time.

The game is confusing because we start with box probabilities, but the game is really about the probabilities between the two groups. There is a 1 in 3 chance the chosen box (group) has the prize and a 2 in 3 chance that the prize is in a box in the unchosen group. The order in which we open the boxes doesn't change either group's chances.

But consider this. Let's say we open the chosen box first and it is empty. Now each box in the unchosen group has a 50/50 chance of having the prize.

The difference in this situation is that we're comparing probabilities of boxes within a group, not between groups.

Here are the latest Covid data. There is a difference in the hospitalization data for US versus NV. The CDC reports total number of Covid hospital admissions. Nevada reports current hospitalizations, not admissions.

All data comes from the CDC (https://www.cdc.gov/covid-data-tracker/#cases_totalcases, https://covid.cdc.gov/covid-data-tracker/#new-hospital-admissions), except for Nevada current hospitalizations (https://nvhealthresponse.nv.gov/).

Totals Weekly Changes
US NV US NV
Date Cases  Hosp. Adm.  Deaths  Cases  Curr. Hosp.  Deaths  Cases  Hosp. Adm.  Deaths  Cases  Curr. Hosp.  Deaths
05/24/22   83,278,995   4,711,767   999,785   731,636   202   10,842   756,047   25,027   2,317   5,324   37   28
05/17/22   82,522,948   4,686,740   997,488   726,312   165   10,814   680,888   21,183   2,097   3,693   35   26
05/10/22   81,842,060   4,665,557   995,371   722,619   130   10,788   534,465   17,934   3,932   2,802   32   11
05/03/22   81,307,595   4,647,623   991,439   719,817   98   10,777   432,666   14,755   2,448   2,213   (6)   49
04/26/22   80,874,929   4,632,868   988,991   717,604   104   10,728   348,507   12,597   2,446   1,447   (22)   46
04/19/22   80,526,422   4,620,271   986,545   716,157   126   10,682   266,330   10,525   3,308   1,232   (37)   497
04/12/22   80,260,092   4,609,746   983,237   714,925   163   10,185   228,465   10,343   3,627   1,228   23   53
04/05/22   80,031,627   4,599,403   979,610   713,697   140   10,132   203,709   10,312   4,105   1,059   (52)   76
03/29/22   79,827,918   4,589,091   975,505   712,638   192   10,056   206,914   9,572   4,083   1,187   5   40
03/22/22   79,621,004   4,579,519   971,422   711,451   187   10,016   200,627   8,502   8,178   742   (71)   83
03/15/22   79,418,377   4,571,017   963,244   710,709   258   9,933   253,069   17,393   5,492   24,844   (20)   113
03/08/22   79,165,308   4,553,624   957,752   685,865   278   9,820   310,308   24,471   9,870   2,099   (101)   116
03/01/22   78,855,000   4,529,153   947,882   683,766   379   9,704   465,845   33,261   14,988   6,421   (189)   196
02/22/22   78,389,155   4,495,892   932,894   677,345   568   9,508   551,462   46,726   12,797   4,023   (297)   173
02/15/22   77,837,693   4,449,166   920,097   673,322   865   9,335   1,005,691   67,963   17,059   5,280   (375)   149
02/08/22   76,782,002   4,381,203   903,038   668,042   1,240   9,186   1,769,556   92,750   18,185   11,144   (387)   235
02/01/22   75,012,446   4,288,453   884,853   656,898   1,627   8,951   3,193,570   121,420   17,885   26,555   (344)   191
01/25/22   71,818,876   4,167,033   866,968   630,343   1,971   8,760   5,102,999   143,329   16,393   57,574   99   133
01/18/22   66,715,937   4,023,704   850,575   572,769   1,872   8,627   4,983,654   147,105   13,301   25,927   (1,787)   99
01/11/22   61,732,283   3,876,599   837,274   546,842   3,659   8,528   5,421,565   139,572   12,168   32,486   2,594   100
01/05/22   56,310,718   3,737,027   825,106   514,344   1,065   8,428   3,501,427   104,272   8,867   20,966   276   61
12/28/21   52,809,291   3,362,755   816,239   493,378   789   8,367   1,693,987   63,185   11,127   7,977   89   56
12/21   51,115,304   3,569,570   805,112   485,401   700   8,311   1,063,296   55,734   9,102   5,639   9   110
12/14   50,052,008   3,513,836   796,010   479,762   691   8,201   853,262   57,885   8,946   5,008   17   91
12/07   49,198,746   3,455,951   787,064   474,754   674   8,110   821,215   66,191   8,575   5,011   23   125
11/30   48,377,531   3,389,760   778,489   469,743   651   7,985   563,746   28,426   6,309   3,343   (50)   55
11/23   47,813,785   3,361,334   772,180   466,400   701   7,930   667,924   41,145   10,754   4,784   (24)   86
11/16   47,145,861   3,320,189   761,426   461,616   725   7,844   604,748   63,164   7,862   4,577   22   85
11/09   46,541,113   3,257,025   753,564   457,039   703   7,759   516,764   33,219   8,290   5,539   34   103
11/02   46,024,349   3,223,806   745,274   451,500   669   7,656   555,915   38,028   9,226   4,327   45   109
10/26   45,468,434   3,185,778   736,048   447,173   624   7,547   488,829   42,265   9,842   4,753   (87)   96
10/19   44,979,605   3,143,513   726,206   442,420   711   7,451   578,396   11,963   4,381   136
10/12   44,401,209   714,243   438,039   7,315   795,586   14,067   15,710   149
10/05   43,605,623   700,176   422,329   7,166   554,194   10,642   3,852   121
09/28   43,051,429   689,534   418,477   7,045   817,218   14,463   6,160   165
09/21   42,234,211   675,071   412,317   6,880   971,637   14,691   7,466   152
09/14   41,262,574   660,380   404,851   6,728   1,176,763   12,919   10,256   145
09/07   40,085,811   647,461   394,595   6,583   975,725   10,076   5,237   104
08/31   39,110,086   637,385   389,358   6,479   1,113,414   9,385   7,592   173
08/24   37,996,672   628,000   381,649   6,306   1,045,491   7,507   8,117   116
08/17   36,951,181   620,493   373,649   6,190   959,978   4,715   7,065   150
08/10   35,991,203   615,778   366,584   6,040   819,524   3,987   7,652   122
08/03   35,171,679   611,791   358,932   5,918   622,832   2,279   7,489   81
07/27   34,548,847   609,012   351,443   5,837   652,251   2,394   8,347   79
07/20   33,896,296   606,618   341,096   5,758   169,933   1,478   3,351   28
07/13   33,726,363   605,140   339,745   5,730   181,047   1,959   4,982   33
07/06   33,545,316   603,181   334,763   5,697   75,104   1,373   2,234   27
06/29   33,470,212   601,808   332,529   5,670   87,507   2,057   3,020   24
06/22   33,382,705   599,751   329,509   5,646   75,420   2,157   1,930   22
06/15   33,303,285   597,594   327,579   5,624   95,797   2,293   1,560   17
06/08   33,207,488   595,301   326,019   5,607   114,250   3,762   2,271   21
06/01   33,039,238   591,539   323,748   5,586   123,333   3,709   991   27
05/25   32,969,905   587,830   322,757   5,559   174,125   4,234   1,676   26
05/18   32,795,780   583,596   321,081   5,533   223,966   4,230   2,301   27
05/11   32,571,814   579,366   318,780   5,506   303,856   4,687   2,541   33
05/04   32,267,958   574,679   316,239   5,473   343,348   4,908   2,559   40
04/27   31,924,610   569,771   313,680   5,433   383,163   4,958   2,747   65
04/20   31,541,447   564,813   310,933   5,368   464,556   5,072   2,590   36
04/13   31,076,891   559,741   308,343   5,332   480,061   5,321   2,986   57
04/06   30,596,830   554,420   305,357   5,275   448,935   7,124   3,084   38
03/30   30,147,895   547,296   302,273   5,237   439,510   6,793   939   63
03/23   29,708,385   540,503   301,334   5,174   388,928   7,446   1,863   53
03/16   29,319,457   533,057   299,471   5,121   381,695   8,362   3,078   81
03/09   28,937,762   524,695   296,393   5,040   480,902   11,573   2,413   83
03/02   28,456,860   513,122   293,980   4,957   463,356   14,129   2,835   75
02/23   27,993,504   498,993   291,145   4,882   451,083   13,923   2,406   162
02/16   27,542,421   485,070   288,739   4,720   602,906   21,411   4,149   198
02/09   26,939,515   463,659   284,590   4,522   779,305   21,828   5,444   244
02/02   26,160,210   441,831   279,146   4,278   1,007,777   22,004   7,249   249
01/26   25,152,433   419,827   271,897   4,029   1,312,565   23,385   10,324   250
01/19   23,839,868   396,442   261,573   3,779   1,317,119   21,318   11,324   279
01/12   22,522,749   375,124   250,249   3,500   1,790,345   22,660   17,217   294
01/05   20,732,404   352,464   233,032   3,206   1,499,561   18,435   14,655   233
John Robison

John Robison is an expert on slot machines and how to play them. John is a slot and video poker columnist and has written for many of gaming’s leading publications. He holds a master's degree in computer science from the prestigious Stevens Institute of Technology.

You may hear John give his slot and video poker tips live on The Good Times Show, hosted by Rudi Schiffer and Mike Schiffer, which is broadcast from Memphis on KXIQ 1180AM Friday afternoon from from 2PM to 5PM Central Time. John is on the show from 4:30 to 5. You can listen to archives of the show on the web anytime.

Books by John Robison:

The Slot Expert's Guide to Playing Slots
John Robison
John Robison is an expert on slot machines and how to play them. John is a slot and video poker columnist and has written for many of gaming’s leading publications. He holds a master's degree in computer science from the prestigious Stevens Institute of Technology.

You may hear John give his slot and video poker tips live on The Good Times Show, hosted by Rudi Schiffer and Mike Schiffer, which is broadcast from Memphis on KXIQ 1180AM Friday afternoon from from 2PM to 5PM Central Time. John is on the show from 4:30 to 5. You can listen to archives of the show on the web anytime.

Books by John Robison:

The Slot Expert's Guide to Playing Slots