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Ask the Wizard #312
"Anonymous" .
According to BookMaker, the Washington Post, who keep a count political false statements in general, would be used as the source of number of lies. According to that source, Trump averaged 15 false statement per day during 2018. The next question to be answered in analyzing this bet is how much time does Trump spend making public statements a day? Between tweets, interviews, and offthecuff statements, 20 minutes seems like a reasonable estimate to me. A nice round number at least. Simple division gives us 15/20 = 0.75 false statements per minute, or one every 80 seconds.
The address was estimated to last six to eight minutes by the media before it started. Let's split the difference and go with seven minutes. Seven minutes at 0.75 false statements a minute gives us an estimated 5.25 false statements. So, I would have set the over/under at 5.5.
By the way, if we assume 5.25 to be the mean number of false statements, then the probability of three or less false statements is 23.17%, if we assume the total is distributed according to the Poisson distribution, which I think is a reasonable assumption.
By the way, in the end, the number of false statements was scored at six.
This question is asked and discussed in the very long thread on Trump at Wizard of Vegas, but discussion of this specific topic starts here.
In singlezero roulette, what is the mean and median number of spins required for every number to appear at least once?
Notnab
Answering the mean is much easier, so we'll start with that. Let's go through it step by step:
 The first spin is definitely going to be a new number.
 The second spin will have a probability of 36/37 of being a new number. If an event has a probability of p, then the expected number of trials for it to occur is 1/p. In this case, the expected number of trials to get the second number is 37/36 = 1.0278.
 After two numbers have been observed, the probability that the next spin will result in a new number is 35/37. Thus, the expected number of spins after the second number to see the third is 37/35 = 1.0571.
 Following this logic, the average number of spins to see every number is 1 + 37/36 + 37/35 + 37/34 + ... + 37/2 + 37/1 = 155.458690.
The median is much more complicated. To find the exact answer, as opposed to using a random simulation, one needs to use a lot of matrix Algebra. I've discussed how to solve similar problems in other Ask the Wizard questions, so I won't go through the details again. One example of a similar question is the one on getting a 66 pair in the hole three times in a row, as discussed in Ask the Wizard #311. Suffice it to say that the probability of seeing every number in 145 spins is 0.49161779, and in 146 spins is 0.501522154. Thus, the median is 146.
This question is asked and discussed in my forum at Wizard of Vegas.
Suppose you have 12 sixsided dice. You roll them and may set aside any dice you wish. You then reroll the other dice. What is the probability of getting a 12ofakind in the two rolls?
"Anonymous" .
There are 58 different types of sequences on the initial roll. The way I identify each is the number of the face in majority, then the number of dice of the face second in total, and so on. For example, a roll of of 3,3,3,3,6,6,6,5,5,2 would be signified as 4321. The following table shows the number of combinations of each sequence, the probability of rolling it, the probability of completing a 12 of a kind in the second roll, and the product of the two. For the probability on the second roll, I assume the player holds the dice that have the greatest total on the initial roll. The lower right cell shows an overall probability of 0.0000037953, which equals 1 in 263,486.
12 Dice Question
Sequence  Combinations  Probability Sequence 
Conditional Probability 
Total Probability 

1200000  6  0.0000000028  1.0000000000  0.0000000028 
1110000  360  0.0000001654  0.1666666667  0.0000000276 
1020000  1,980  0.0000009096  0.0277777778  0.0000000253 
1011000  7,920  0.0000036384  0.0277777778  0.0000001011 
930000  6,600  0.0000030320  0.0046296296  0.0000000140 
921000  79,200  0.0000363840  0.0046296296  0.0000001684 
911100  79,200  0.0000363840  0.0046296296  0.0000001684 
840000  14,850  0.0000068220  0.0007716049  0.0000000053 
831000  237,600  0.0001091519  0.0007716049  0.0000000842 
822000  178,200  0.0000818639  0.0007716049  0.0000000632 
821100  1,069,200  0.0004911837  0.0007716049  0.0000003790 
811110  356,400  0.0001637279  0.0007716049  0.0000001263 
750000  23,760  0.0000109152  0.0001286008  0.0000000014 
741000  475,200  0.0002183039  0.0001286008  0.0000000281 
732000  950,400  0.0004366077  0.0001286008  0.0000000561 
731100  2,851,200  0.0013098232  0.0001286008  0.0000001684 
722100  4,276,800  0.0019647348  0.0001286008  0.0000002527 
721110  5,702,400  0.0026196464  0.0001286008  0.0000003369 
711111  570,240  0.0002619646  0.0001286008  0.0000000337 
660000  13,860  0.0000063672  0.0000214335  0.0000000001 
651000  665,280  0.0003056254  0.0000214335  0.0000000066 
642000  1,663,200  0.0007640635  0.0000214335  0.0000000164 
641100  4,989,600  0.0022921906  0.0000214335  0.0000000491 
633000  1,108,800  0.0005093757  0.0000214335  0.0000000109 
632100  19,958,400  0.0091687624  0.0000214335  0.0000001965 
631110  13,305,600  0.0061125083  0.0000214335  0.0000001310 
622200  4,989,600  0.0022921906  0.0000214335  0.0000000491 
622110  29,937,600  0.0137531436  0.0000214335  0.0000002948 
621111  9,979,200  0.0045843812  0.0000214335  0.0000000983 
552000  997,920  0.0004584381  0.0000035722  0.0000000016 
551100  2,993,760  0.0013753144  0.0000035722  0.0000000049 
543000  3,326,400  0.0015281271  0.0000035722  0.0000000055 
542100  29,937,600  0.0137531436  0.0000035722  0.0000000491 
541110  19,958,400  0.0091687624  0.0000035722  0.0000000328 
533100  19,958,400  0.0091687624  0.0000035722  0.0000000328 
532200  29,937,600  0.0137531436  0.0000035722  0.0000000491 
532110  119,750,400  0.0550125743  0.0000035722  0.0000001965 
531111  19,958,400  0.0091687624  0.0000035722  0.0000000328 
52221  59,875,200  0.0275062872  0.0000035722  0.0000000983 
522111  59,875,200  0.0275062872  0.0000035722  0.0000000983 
444000  693,000  0.0003183598  0.0000005954  0.0000000002 
443100  24,948,000  0.0114609530  0.0000005954  0.0000000068 
442200  18,711,000  0.0085957147  0.0000005954  0.0000000051 
442110  74,844,000  0.0343828589  0.0000005954  0.0000000205 
441111  12,474,000  0.0057304765  0.0000005954  0.0000000034 
433200  49,896,000  0.0229219060  0.0000005954  0.0000000136 
433110  99,792,000  0.0458438119  0.0000005954  0.0000000273 
432210  299,376,000  0.1375314358  0.0000005954  0.0000000819 
432111  199,584,000  0.0916876238  0.0000005954  0.0000000546 
422220  37,422,000  0.0171914295  0.0000005954  0.0000000102 
422211  149,688,000  0.0687657179  0.0000005954  0.0000000409 
333300  5,544,000  0.0025468784  0.0000000992  0.0000000003 
333210  133,056,000  0.0611250826  0.0000000992  0.0000000061 
333111  44,352,000  0.0203750275  0.0000000992  0.0000000020 
332220  99,792,000  0.0458438119  0.0000000992  0.0000000045 
332211  299,376,000  0.1375314358  0.0000000992  0.0000000136 
322221  149,688,000  0.0687657179  0.0000000992  0.0000000068 
222222  7,484,400  0.0034382859  0.0000000165  0.0000000001 
Total  2,176,782,336  1.0000000000  0.0000037953 
Sam from Fountain Valley CA
As measured by square feet of gambling space, here they are. This comes as a surprise to me, as I've barely heard of the two Oklahoma casinos in the top five.
Top Five U.S. Casinos
Casino  Location  Square Feet 

Winstar  Thackerville, OK  519,000 
Mohegan Sun  Uncasville, Connecticut  364,000 
Foxwoods  Mashantucket, CT  344,000 
San Manuel  Highland CA  220,000 
Riverwind  Norman OK  216,000 