Question

The probability of throwing “box cars” (two sixes) with a pair of dice is 1/62. Show this using Monte Carlo method and determine the probability of throwing box cars twice in a row.

Answer 1

The probability of throwing "box cars" (two sixes ) is not 1/62 but 1/36 because there are a total of 6*6 - 36 combinations in throw of 2 dice and only one combination is 66.

This can be proved using monte carlo method as:

S.No	Dice 1	Dice 2	Both Sixes
1	5	4	0
2	5	2	0
3	2	3	0
4	1	5	0
5	1	6	0
6	2	4	0
7	1	6	0
8	5	6	0
9	6	3	0
10	6	6	1
11	4	2	0
12	6	5	0
13	4	4	0
14	2	2	0
15	4	3	0
16	3	1	0
17	6	4	0
18	2	5	0
19	3	5	0
20	6	5	0
21	6	5	0
22	3	4	0
23	2	5	0
24	2	1	0
25	2	3	0
26	6	2	0
27	1	1	0
28	3	3	0
29	3	2	0
30	3	2	0
31	5	5	0
32	3	5	0
33	4	6	0
34	6	5	0
35	6	2	0
36	3	1	0
37	3	5	0
38	6	2	0
39	1	4	0
40	6	4	0
41	4	1	0
42	2	4	0
43	3	3	0
44	6	3	0
45	2	5	0
46	4	4	0
47	1	4	0
48	3	5	0
49	6	1	0
50	6	2	0
51	4	3	0
52	6	5	0
53	4	3	0
54	6	4	0
55	5	4	0
56	3	5	0
57	4	1	0
58	4	1	0
59	1	2	0
60	3	3	0
61	1	4	0
62	6	6	1
63	4	6	0
64	3	6	0
65	6	1	0
66	3	2	0
67	5	2	0
68	1	2	0
69	1	6	0
70	2	5	0
71	1	5	0
72	2	4	0
73	5	4	0
74	6	2	0
75	5	2	0
76	1	6	0
77	2	1	0
78	1	6	0
79	5	4	0
80	1	5	0
81	2	4	0
82	4	1	0
83	2	5	0
84	4	5	0
85	4	4	0
86	6	5	0
87	6	3	0
88	6	4	0
89	6	5	0
90	2	3	0
91	5	5	0
92	2	4	0
93	2	5	0
94	2	3	0
95	1	4	0
96	1	1	0
97	4	3	0
98	6	2	0
99	5	1	0
100	2	4	0
101	2	2	0
102	5	3	0
103	6	2	0
104	6	4	0
105	3	4	0
106	2	2	0
107	3	5	0
108	2	5	0
109	3	2	0
110	3	5	0
111	6	6	1
112	5	1	0
113	3	2	0
114	2	2	0
115	1	6	0
116	5	3	0
117	1	2	0
118	1	1	0
119	4	6	0
120	3	5	0
121	2	2	0
122	5	3	0
123	2	1	0
124	2	4	0
125	5	6	0
126	5	2	0
127	5	5	0
128	6	3	0
129	1	5	0
130	2	1	0
131	2	1	0
132	6	6	1
133	1	2	0
134	3	1	0
135	1	3	0
136	2	5	0
137	4	4	0
138	6	2	0
139	4	5	0
140	2	3	0
141	3	6	0
142	6	5	0
143	2	6	0
144	2	3	0
			4

We produced 144 simulations for 2 dice throws, we see that only in 4 out of those 144 cases, we are getting 2 sixes, therefore from monte carlo method, the required probability here has come out to be:

= 4/144

= 1/36 which was the initial hypothesis probability we came out with.

Now the probability of getting 2 pairs of sixes in a row is computed as:

= (1/36)²

= 1/1296

Therefore 1/1296 is the required probability here.