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Ask the Wizard #340

If a casino increased the win on the Tie bet to 9 to 1, above the usual 8 to 1, how much additional wagering would it need on the Tie to have the same expected win?

dandolos2000

The probability of a tie in baccarat is 0.095155968.

At the usual win of 8 to 1, the expected return to the player is 0.095156 × (8+1) - 1 = -0.143596.

At the a win of 9 to 1, the expected return to the player is 0.095156 × (9+1) - 1 = --0.048440.

The expected player loss is 0.143596/0.048440 = 2.9643960 times higher at a win of 8 to 1. Thus, the casino would need 2.9643960 times as much action on the Tie if they increased the win to 9 to 1 for the expected casino win to be the same.

This question is raised and discussed in my forum at Wizard of Vegas.

Suppose there is a bin with 100 balls, numbered 1 to 100. Ten balls are drawn at random, without replacement. What is the mean number of the lowest ball drawn?

ThatDonGuy

The following table shows the number of combinations, probability, and contribution to the lowest ball (product of ball and probability). The lower right cell shows the expected lowest ball is 9.1818182.

Lowest Ball

Lowest
Ball
Combinations Probability Expected
Low Ball
1 1,731,030,945,644 0.100000 0.100000
2 1,573,664,496,040 0.090909 0.181818
3 1,429,144,287,220 0.082560 0.247681
4 1,296,543,270,880 0.074900 0.299600
5 1,174,992,339,235 0.067878 0.339391
6 1,063,677,275,518 0.061448 0.368686
7 961,835,834,245 0.055564 0.388950
8 868,754,947,060 0.050187 0.401497
9 783,768,050,065 0.045278 0.407498
10 706,252,528,630 0.040800 0.407995
11 635,627,275,767 0.036720 0.403915
12 571,350,360,240 0.033006 0.396076
13 512,916,800,670 0.029631 0.385199
14 459,856,441,980 0.026565 0.371917
15 411,731,930,610 0.023785 0.356780
16 368,136,785,016 0.021267 0.340271
17 328,693,558,050 0.018988 0.322801
18 293,052,087,900 0.016929 0.304728
19 260,887,834,350 0.015071 0.286354
20 231,900,297,200 0.013397 0.267933
21 205,811,513,765 0.011890 0.249680
22 182,364,632,450 0.010535 0.231771
23 161,322,559,475 0.009319 0.214347
24 142,466,675,900 0.008230 0.197524
25 125,595,622,175 0.007256 0.181388
26 110,524,147,514 0.006385 0.166007
27 97,082,021,465 0.005608 0.151425
28 85,113,005,120 0.004917 0.137673
29 74,473,879,480 0.004302 0.124766
30 65,033,528,560 0.003757 0.112708
31 56,672,074,888 0.003274 0.101491
32 49,280,065,120 0.002847 0.091100
33 42,757,703,560 0.002470 0.081512
34 37,014,131,440 0.002138 0.072701
35 31,966,749,880 0.001847 0.064634
36 27,540,584,512 0.001591 0.057276
37 23,667,689,815 0.001367 0.050589
38 20,286,591,270 0.001172 0.044534
39 17,341,763,505 0.001002 0.039071
40 14,783,142,660 0.000854 0.034160
41 12,565,671,261 0.000726 0.029762
42 10,648,873,950 0.000615 0.025837
43 8,996,462,475 0.000520 0.022348
44 7,575,968,400 0.000438 0.019257
45 6,358,402,050 0.000367 0.016529
46 5,317,936,260 0.000307 0.014132
47 4,431,613,550 0.000256 0.012032
48 3,679,075,400 0.000213 0.010202
49 3,042,312,350 0.000176 0.008612
50 2,505,433,700 0.000145 0.007237
51 2,054,455,634 0.000119 0.006053
52 1,677,106,640 0.000097 0.005038
53 1,362,649,145 0.000079 0.004172
54 1,101,716,330 0.000064 0.003437
55 886,163,135 0.000051 0.002816
56 708,930,508 0.000041 0.002293
57 563,921,995 0.000033 0.001857
58 445,891,810 0.000026 0.001494
59 350,343,565 0.000020 0.001194
60 273,438,880 0.000016 0.000948
61 211,915,132 0.000012 0.000747
62 163,011,640 0.000009 0.000584
63 124,403,620 0.000007 0.000453
64 94,143,280 0.000005 0.000348
65 70,607,460 0.000004 0.000265
66 52,451,256 0.000003 0.000200
67 38,567,100 0.000002 0.000149
68 28,048,800 0.000002 0.000110
69 20,160,075 0.000001 0.000080
70 14,307,150 0.000001 0.000058
71 10,015,005 0.000001 0.000041
72 6,906,900 0.000000 0.000029
73 4,686,825 0.000000 0.000020
74 3,124,550 0.000000 0.000013
75 2,042,975 0.000000 0.000009
76 1,307,504 0.000000 0.000006
77 817,190 0.000000 0.000004
78 497,420 0.000000 0.000002
79 293,930 0.000000 0.000001
80 167,960 0.000000 0.000001
81 92,378 0.000000 0.000000
82 48,620 0.000000 0.000000
83 24,310 0.000000 0.000000
84 11,440 0.000000 0.000000
85 5,005 0.000000 0.000000
86 2,002 0.000000 0.000000
87 715 0.000000 0.000000
88 220 0.000000 0.000000
89 55 0.000000 0.000000
90 10 0.000000 0.000000
91 1 0.000000 0.000000
Total 17,310,309,456,440 1.000000 9.181818

There is an easier way to solve problems like this, where the lowest ball is 1. The formula for the lowest ball is (m+1)/(b+1), where m is the maximum ball value and b is the number of balls. In this case, m=100 and n=10, so the lowest ball is 101/11 = 9.181818.

This question is asked and discussed in my forum at Wizard of Vegas.

The following puzzle appeared in the March 6, 2021 New York Times.

The rules are pretty simple:

  1. Each row, column and region must have exactly two stars.
  2. No two stars may touch, not even diagonally.

Can you help with a solution?

"Anonymous" .

This is called a Two not Touch puzzle. The button below shows my answer and solution.

two not touch solved

Here is my solution (PDF).

What is the expected number of rolls of a fair six-side die for any one side to be rolled six times?

Ace2

Click the button below for my answer.

The answer is 2597868106693535971 / 131621703842267136 = Approximation: 19.73738396371749

Here is my solution (PDF).