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# Ask the Slot Expert: Monty Hall plays Survivor

18 May 2022

The current and prior seasons of Survivor added a twist that illustrates the Monty Hall Problem.

Survivor has two types of challenges. In an endurance or last-man-standing challenge, the last person still standing on a buoy or balancing a ball on a plank wins. In a skill or finish-first challenge, the first person to complete a task or solve a puzzle wins.

The new twist in the past two seasons is forcing the survivor who finishes last in a skill challenge to play a game called Do or Die at the next tribal council. The survivor is shown three boxes. Inside one box is the image of a flame, which means the survivor is safe that night and cannot be voted out. The other two boxes contain an image of a skull, which means that the survivor is eliminated from the game.

This is the classic Let's Make a Deal situation. You have a choice of three boxes or doors. Monty (or, more recently, Wayne Brady) tells you that behind one door is a car or some other fabulous prize and behind the other two are zonks, like goats.

You make your choice and then the hosts reveals the zonk or skull inside one of the two items you didn't choose. The host then gives you the option to stick with your original choice or switch to the remaining item.

Should you switch? (Well, what do you think? Should you?)

Both times the game has been played on Survivor, the survivors stayed with their original choice and both times the boxes contained the flame and the survivor was safe.

Mathematically speaking, the players made a poor choice. They should have switched.

There are three choices. One is good and two are bad. The probability of choosing the good outcome is 1/3 and the probability of choosing the bad is 2/3.

You make your choice. There is a 1/3 probability that you have the good box. Each of the other two boxes also has a 1/3 probability of being good, so there's a 2/3 chance the good box is one of the boxes you did not choose. We also know that at least of of the boxes you didn't choose is bad.

The boxes are partitioned into two groups: one containing one box with a 1/3 chance of being good, and another containing two boxes with a combined 2/3 chance of being good.

Jeff (or Monty or Wayne) reveals one of the boxes you didn't choose. It is bad. That is not new information. We already knew that at least one of the boxes you didn't choose was bad.

Some people may think that the probability that their box is good has increased to 50%, but it's still 1/3 because we were not given any new information. What has happened, though, is the probability that the remaining box you did not pick is now 2/3.

You have a 1/3 chance of having chosen the good box and the other box has a 2/3 chance of being good. That's why you should switch.

Let's call the box you chose Box 1 and the other two Box 2 and Box 3. The following table shows what could be in each box and what happens if you stay with your original pick or switch.

Box 1Box 2Box 3StaySwitch
GBBWinLose
BGBLoseWin
BBGLoseWin

You win 2/3 of the time when you switch and only 1/3 of the time when you stay. If only we could know when it's that 1/3 time when we should stick with our original choice. Because we don't know, we play the probabilities and switch.

Here's another way to look at it. If you're going to stay with your original choice, the only way you win is if you chose the good box and you had a 1/3 chance of doing so. The other 2/3 is with the other two boxes and the host eliminated one of the boxes for you.

The outcomes of the games played on Survivor illustrate a consequence of statistical decision making. Sometimes bad decisions lead to good outcomes. Conversely, sometimes good decisions lead to bad outcomes.

The game also illustrates an aspect of probability that many people don't understand. Here's a transcript of a segment from the November 6, 2020 episode of Tooning Out the News. In the segment they played a clip from ABC's election night coverage in which George Stephanopoulos talks to pollster Nate Silver.

GS: Let's bring in Nate Silver from FiveThirtyEight. Nate, talk about the forecast you had coming into election night.

NS: So, coming in we had Joe Biden with an 89% chance of winning, Trump with 10%, 1% chance of a tie.  So in 2016 we had Trump with a 30% chance of winning, roughly....  But look, you know, I don't control when the 30% happens, the 10% happens, so there--

At this point, an animated reporter walks into frame and puts an animated sash that says Mr Wrong 2020 on Mr. Silver.

Reporter: Congratulations to Nate Silver, the new Mr. Wrong 2020. As Mr. Wrong, you will write a think piece about why you were actually right before the news media gives you another chance in 2024.

Thirty percent, 10 percent, 1 percent, whatever, are not 0 percent. You have about a 0.0025 percent chance of hitting a royal flush, yet we all hope to hit one and 99.999 percent of us have hit one or more.

Gamblers understand that even a small chance is better than no chance and they never know when an event will occur. Probability may say that you're better off switching, but sometimes you win when you don't. Nate Silver was not wrong.

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
05/17/22   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