Keeping Track of Coin Flips Even When They Are Not Flipped

Question

I am working with the R programming language.

I have the following problem:

• There are 100 coins: the coins are randomly initialized at turn=0 (i.e. 0.5 probability of heads and 0.5 probability of tails)
• At each turn, any given coin has a 0.5 probability of being selected and flipped
• When a coin is flipped, there is a 0.6 probability it will land on the same side that was facing upwards when flipped and a 0.4 probability of landing on the other side
• At the end of each turn, we check each coin (flipped or not flipped, does not matter). If a coin is heads up then +1 else -1
• We do this for 100 turns

My Question: For each of these 100 turns, for each of these 100 coins - I want to track the cumulative number of heads and the cumulative number of tails.

As an example, suppose coin43 was flipped 7 times:

• heads, tails, not selected, not selected, heads, not selected, tails

Then, the scores would be:

• +1 , -1, -1, -1, +1, +1, -1

And at the most recent turn, the cumulative numbers would be:

• 3 heads and 4 tails

First, I initialized the coins:

num_coins <- 100
num_turns <- 100
coins <- sample(c(-1, 1), num_coins, replace = TRUE)
cumulative_heads <- matrix(0, nrow = num_turns, ncol = num_coins)
cumulative_tails <- matrix(0, nrow = num_turns, ncol = num_coins)

Next, I tried to write the bulk of the simulation code:

# Simulation
for (turn in 1:num_turns) {
  if(turn > 1){

    cumulative_heads[turn,] <- cumulative_heads[turn-1,]
    cumulative_tails[turn,] <- cumulative_tails[turn-1,]
  }
  for (coin in 1:num_coins) {
    # Check if coin is selected
    if (runif(1) < 0.5) {
      # Flip the coin
      if (runif(1) < 0.6) {
        # Coin lands on the same side
        coins[coin] <- coins[coin]
      } else {
        # Coin lands on the other side
        coins[coin] <- -coins[coin]
      }
    }
    # Update cumulative counts
    if (coins[coin] == 1) {
      cumulative_heads[turn, coin] <- cumulative_heads[turn, coin] + 1
    } else {
      cumulative_tails[turn, coin] <- cumulative_tails[turn, coin] + 1
    }
  }
}

Then, I created a data frame to store the results:

results <- data.frame(matrix(ncol = num_coins, nrow = num_turns))
names(results) <- paste0("coin", 1:num_coins)

for (turn in 1:num_turns) {
  for (coin in 1:num_coins) {
    results[turn, coin] <- paste("Heads: ", cumulative_heads[turn, coin], ", Tails: ", cumulative_tails[turn, coin])
  }
}

The final results look something like this (a sample):

> results[1:5, 1:5]
                  coin1                 coin2                 coin3                 coin4                 coin5
1 Heads:  1 , Tails:  0 Heads:  0 , Tails:  1 Heads:  1 , Tails:  0 Heads:  1 , Tails:  0 Heads:  1 , Tails:  0
2 Heads:  1 , Tails:  1 Heads:  0 , Tails:  2 Heads:  2 , Tails:  0 Heads:  2 , Tails:  0 Heads:  2 , Tails:  0
3 Heads:  1 , Tails:  2 Heads:  0 , Tails:  3 Heads:  3 , Tails:  0 Heads:  3 , Tails:  0 Heads:  2 , Tails:  1
4 Heads:  2 , Tails:  2 Heads:  0 , Tails:  4 Heads:  4 , Tails:  0 Heads:  4 , Tails:  0 Heads:  2 , Tails:  2
5 Heads:  3 , Tails:  2 Heads:  0 , Tails:  5 Heads:  5 , Tails:  0 Heads:  5 , Tails:  0 Heads:  2 , Tails:  3

I think I have overcomplicated this - can someone please show me what I can do to simplify this?

Thanks!

Answer 1

Idea and Code

You can define a function that updates the selected coins for flips, e.g.,

f <- function(v) {
   selected <- runif(length(v)) <= 0.5
   updated <- v[selected] * (1 - 2 * (runif(sum(selected)) <= 0.6))
   replace(v, selected, updated)
}

then we use Reduce function (but enable the accumulate argument) to keep track of the progress of changes, where the updated coin status from the previous iteration will be the taken as the input for the upcoming iteration

set.seed(0)
num_coins <- 100
num_turns <- 100
v <- sample(c(1, -1), num_coins, TRUE, prob = c(0.5, 0.5))
p <- Reduce(\(x, y) f(x), seq.int(num_turns), init = v, accumulate = TRUE)

Finally, if you would like to obtain the cumulative statistics of the heads and tails of each coin, you can first rbind the outcomes of p, and then analyze the distribution of heads and tails by column, e.g.,

out <- apply(
   do.call(rbind, p),
   2,
   \(u) data.frame(
      heads = cumsum(u == 1),
      tails = cumsum(u == -1)
   )
)

Output

The outcome of the first 6 turns (incl. turn=0) is shown as below

> head(p)
[[1]]
  [1]  1 -1 -1  1  1 -1  1  1  1  1 -1 -1 -1  1 -1  1 -1  1  1 -1  1  1 -1  1 -1
 [26] -1 -1 -1 -1  1 -1 -1  1 -1 -1  1  1  1 -1  1 -1  1  1  1  1  1  1 -1 -1  1
 [51]  1 -1  1 -1 -1 -1 -1 -1  1  1 -1  1 -1 -1 -1  1 -1 -1  1 -1  1 -1  1 -1 -1
 [76] -1  1  1 -1  1  1 -1  1 -1 -1  1 -1  1 -1 -1 -1 -1 -1  1  1  1  1 -1 -1  1

[[2]]
  [1]  1 -1 -1 -1  1 -1 -1 -1 -1  1 -1 -1 -1 -1  1 -1  1  1 -1 -1  1  1 -1 -1 -1
 [26] -1 -1 -1 -1 -1 -1 -1  1 -1 -1  1  1  1 -1  1 -1  1  1  1 -1  1 -1  1 -1 -1
 [51]  1 -1  1  1 -1 -1  1 -1 -1 -1  1  1 -1  1 -1  1  1  1  1 -1 -1 -1  1 -1 -1
 [76] -1  1  1 -1  1  1  1 -1 -1 -1  1  1  1 -1 -1 -1 -1  1 -1  1  1  1  1 -1 -1

[[3]]
  [1] -1  1  1  1  1 -1 -1 -1 -1 -1  1 -1  1 -1  1 -1 -1  1 -1 -1 -1  1 -1 -1  1
 [26]  1  1 -1  1  1 -1  1  1 -1 -1 -1  1  1  1 -1 -1 -1 -1 -1  1  1  1  1  1 -1
 [51]  1  1 -1  1 -1 -1  1  1  1 -1  1  1 -1 -1 -1 -1 -1  1  1 -1  1  1 -1 -1 -1
 [76] -1  1 -1 -1  1  1  1 -1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1  1  1  1  1 -1

[[4]]
  [1] -1  1  1  1  1 -1 -1 -1  1 -1 -1 -1  1  1  1 -1 -1  1  1 -1 -1  1 -1 -1 -1
 [26] -1  1 -1  1  1 -1  1  1  1  1 -1  1 -1  1 -1 -1 -1 -1 -1  1  1 -1 -1  1 -1
 [51]  1  1  1 -1  1  1 -1 -1  1 -1  1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1  1 -1 -1  1
 [76] -1  1 -1 -1 -1  1  1 -1 -1 -1 -1 -1  1  1 -1 -1 -1 -1 -1  1  1  1  1  1 -1

[[5]]
  [1] -1 -1  1  1  1 -1  1 -1  1 -1  1  1  1 -1  1 -1 -1 -1  1 -1 -1  1 -1 -1 -1
 [26]  1  1 -1  1  1 -1 -1  1  1  1 -1 -1  1 -1  1 -1 -1  1 -1  1  1  1 -1  1 -1
 [51]  1 -1  1 -1  1 -1  1  1  1  1  1  1 -1  1 -1 -1  1 -1  1 -1 -1  1  1 -1 -1
 [76]  1  1  1 -1 -1  1  1 -1  1 -1 -1 -1  1  1 -1 -1 -1 -1 -1  1  1 -1  1  1 -1

[[6]]
  [1]  1 -1  1  1  1 -1 -1 -1 -1  1 -1 -1  1 -1 -1 -1 -1 -1 -1 -1 -1  1 -1  1 -1
 [26]  1  1 -1  1 -1  1 -1  1  1  1 -1 -1 -1  1 -1  1 -1  1 -1  1  1  1 -1  1 -1
 [51] -1  1  1  1  1 -1  1  1 -1  1  1  1 -1  1  1 -1 -1 -1  1  1  1  1 -1 -1 -1
 [76]  1  1 -1 -1 -1  1  1  1  1 -1  1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1  1 -1

and the updating progress of the first 6 coins

> head(out)
[[1]]
    heads tails
1       1     0
2       2     0
3       2     1
4       2     2
5       2     3
6       3     3
7       3     4
8       3     5
9       3     6
10      3     7
11      4     7
12      5     7
13      6     7
14      7     7
15      7     8
16      8     8
17      9     8
18     10     8
19     11     8
20     12     8
21     12     9
22     12    10
23     12    11
24     12    12
25     12    13
26     12    14
27     12    15
28     12    16
29     12    17
30     13    17
31     13    18
32     14    18
33     15    18
34     16    18
35     17    18
36     17    19
37     17    20
38     17    21
39     17    22
40     17    23
41     17    24
42     17    25
43     17    26
44     18    26
45     19    26
46     20    26
47     21    26
48     22    26
49     23    26
50     23    27
51     23    28
52     23    29
53     24    29
54     24    30
55     24    31
56     24    32
57     24    33
58     24    34
59     24    35
60     24    36
61     24    37
62     24    38
63     25    38
64     26    38
65     27    38
66     28    38
67     29    38
68     30    38
69     31    38
70     32    38
71     33    38
72     33    39
73     33    40
74     33    41
75     33    42
76     33    43
77     33    44
78     34    44
79     35    44
80     36    44
81     37    44
82     37    45
83     37    46
84     37    47
85     38    47
86     39    47
87     40    47
88     41    47
89     41    48
90     41    49
91     41    50
92     41    51
93     42    51
94     43    51
95     44    51
96     44    52
97     44    53
98     44    54
99     44    55
100    44    56
101    45    56

[[2]]
    heads tails
1       0     1
2       0     2
3       1     2
4       2     2
5       2     3
6       2     4
7       3     4
8       4     4
9       4     5
10      4     6
11      4     7
12      4     8
13      4     9
14      4    10
15      4    11
16      4    12
17      5    12
18      6    12
19      6    13
20      6    14
21      7    14
22      7    15
23      7    16
24      7    17
25      7    18
26      7    19
27      7    20
28      7    21
29      7    22
30      7    23
31      8    23
32      9    23
33     10    23
34     10    24
35     11    24
36     12    24
37     13    24
38     14    24
39     15    24
40     16    24
41     17    24
42     18    24
43     18    25
44     18    26
45     18    27
46     18    28
47     18    29
48     18    30
49     19    30
50     19    31
51     20    31
52     21    31
53     21    32
54     21    33
55     21    34
56     21    35
57     21    36
58     21    37
59     21    38
60     22    38
61     23    38
62     24    38
63     24    39
64     24    40
65     24    41
66     25    41
67     26    41
68     27    41
69     28    41
70     29    41
71     29    42
72     29    43
73     29    44
74     29    45
75     30    45
76     30    46
77     30    47
78     31    47
79     32    47
80     33    47
81     34    47
82     35    47
83     36    47
84     37    47
85     37    48
86     37    49
87     37    50
88     37    51
89     37    52
90     38    52
91     39    52
92     40    52
93     41    52
94     42    52
95     43    52
96     43    53
97     44    53
98     45    53
99     46    53
100    47    53
101    48    53

[[3]]
    heads tails
1       0     1
2       0     2
3       1     2
4       2     2
5       3     2
6       4     2
7       4     3
8       4     4
9       4     5
10      4     6
11      4     7
12      5     7
13      5     8
14      5     9
15      5    10
16      6    10
17      7    10
18      8    10
19      9    10
20     10    10
21     11    10
22     12    10
23     13    10
24     14    10
25     15    10
26     16    10
27     16    11
28     16    12
29     17    12
30     18    12
31     19    12
32     19    13
33     19    14
34     19    15
35     19    16
36     20    16
37     21    16
38     22    16
39     23    16
40     24    16
41     25    16
42     25    17
43     26    17
44     27    17
45     28    17
46     28    18
47     28    19
48     28    20
49     29    20
50     30    20
51     31    20
52     32    20
53     32    21
54     32    22
55     32    23
56     32    24
57     32    25
58     33    25
59     34    25
60     35    25
61     36    25
62     37    25
63     38    25
64     39    25
65     40    25
66     40    26
67     40    27
68     40    28
69     41    28
70     41    29
71     42    29
72     43    29
73     44    29
74     45    29
75     46    29
76     47    29
77     48    29
78     49    29
79     50    29
80     51    29
81     51    30
82     51    31
83     51    32
84     52    32
85     53    32
86     53    33
87     54    33
88     54    34
89     55    34
90     56    34
91     57    34
92     58    34
93     59    34
94     60    34
95     61    34
96     61    35
97     61    36
98     61    37
99     62    37
100    63    37
101    64    37

[[4]]
    heads tails
1       1     0
2       1     1
3       2     1
4       3     1
5       4     1
6       5     1
7       5     2
8       6     2
9       7     2
10      7     3
11      8     3
12      9     3
13      9     4
14     10     4
15     10     5
16     10     6
17     10     7
18     11     7
19     11     8
20     11     9
21     11    10
22     11    11
23     11    12
24     12    12
25     13    12
26     13    13
27     13    14
28     13    15
29     13    16
30     13    17
31     14    17
32     15    17
33     16    17
34     16    18
35     16    19
36     16    20
37     16    21
38     17    21
39     18    21
40     18    22
41     18    23
42     18    24
43     18    25
44     18    26
45     18    27
46     18    28
47     18    29
48     18    30
49     18    31
50     18    32
51     18    33
52     18    34
53     18    35
54     18    36
55     19    36
56     20    36
57     21    36
58     21    37
59     21    38
60     21    39
61     21    40
62     21    41
63     22    41
64     23    41
65     24    41
66     25    41
67     26    41
68     27    41
69     27    42
70     28    42
71     29    42
72     30    42
73     31    42
74     32    42
75     32    43
76     32    44
77     32    45
78     32    46
79     33    46
80     34    46
81     35    46
82     36    46
83     36    47
84     36    48
85     37    48
86     38    48
87     38    49
88     39    49
89     40    49
90     41    49
91     41    50
92     42    50
93     43    50
94     44    50
95     45    50
96     46    50
97     46    51
98     47    51
99     47    52
100    47    53
101    47    54

[[5]]
    heads tails
1       1     0
2       2     0
3       3     0
4       4     0
5       5     0
6       6     0
7       7     0
8       7     1
9       8     1
10      9     1
11     10     1
12     11     1
13     12     1
14     12     2
15     13     2
16     14     2
17     14     3
18     14     4
19     14     5
20     15     5
21     15     6
22     15     7
23     15     8
24     15     9
25     15    10
26     15    11
27     15    12
28     16    12
29     16    13
30     16    14
31     16    15
32     16    16
33     16    17
34     16    18
35     17    18
36     18    18
37     19    18
38     20    18
39     21    18
40     22    18
41     23    18
42     24    18
43     25    18
44     26    18
45     27    18
46     27    19
47     27    20
48     27    21
49     28    21
50     28    22
51     29    22
52     30    22
53     30    23
54     31    23
55     31    24
56     31    25
57     31    26
58     31    27
59     31    28
60     32    28
61     33    28
62     33    29
63     33    30
64     34    30
65     34    31
66     34    32
67     34    33
68     35    33
69     36    33
70     37    33
71     38    33
72     39    33
73     40    33
74     41    33
75     42    33
76     43    33
77     44    33
78     44    34
79     44    35
80     44    36
81     44    37
82     44    38
83     44    39
84     44    40
85     44    41
86     45    41
87     45    42
88     45    43
89     46    43
90     47    43
91     48    43
92     48    44
93     48    45
94     48    46
95     49    46
96     50    46
97     51    46
98     51    47
99     51    48
100    52    48
101    53    48

[[6]]
    heads tails
1       0     1
2       0     2
3       0     3
4       0     4
5       0     5
6       0     6
7       1     6
8       2     6
9       2     7
10      3     7
11      3     8
12      3     9
13      3    10
14      3    11
15      3    12
16      3    13
17      3    14
18      4    14
19      4    15
20      4    16
21      4    17
22      4    18
23      4    19
24      5    19
25      6    19
26      6    20
27      7    20
28      8    20
29      8    21
30      8    22
31      9    22
32     10    22
33     11    22
34     11    23
35     11    24
36     11    25
37     12    25
38     12    26
39     12    27
40     12    28
41     12    29
42     12    30
43     12    31
44     12    32
45     12    33
46     13    33
47     14    33
48     14    34
49     14    35
50     15    35
51     16    35
52     17    35
53     17    36
54     17    37
55     17    38
56     17    39
57     18    39
58     19    39
59     20    39
60     20    40
61     20    41
62     21    41
63     22    41
64     23    41
65     23    42
66     24    42
67     24    43
68     25    43
69     26    43
70     27    43
71     28    43
72     29    43
73     29    44
74     29    45
75     29    46
76     29    47
77     29    48
78     29    49
79     29    50
80     29    51
81     29    52
82     29    53
83     30    53
84     31    53
85     32    53
86     32    54
87     32    55
88     32    56
89     32    57
90     32    58
91     32    59
92     33    59
93     34    59
94     35    59
95     36    59
96     37    59
97     38    59
98     39    59
99     40    59
100    40    60
101    41    60

Answer 2

Break the code into its several steps.

Step 1
Flip the coins

1. Flip one coin one turn;
2. Flip one coin n turns;
3. Flip all coins all turns.

flip_one_coin <- function() {
  if (runif(1) < 0.5) {
    # Flip the coin
    if (runif(1) < 0.6) {
      1L      # Coin lands on the same side
    } else {
      -1L     # Coin lands on the other side
    }
  } else -1L  # Coin not selected
}
flip_one_coin_ntimes <- function(n) {
  heads <- tails <- replicate(n, flip_one_coin())
  heads[heads != 1L] <- 0L
  tails[tails != -1L] <- 0L
  list(Heads = cumsum(heads), Tails = -cumsum(tails))
}
flip_all_coins <- function(ncoins, ntimes) {
  replicate(ncoins, flip_one_coin_ntimes(ntimes))
}

num_coins <- 100L
num_turns <- 100L

# test the function and see what it outputs
flip_one_coin_ntimes(num_turns)
#> $Heads
#>   [1]  0  0  0  1  2  2  2  2  2  2  2  2  2  3  3  3  3  4  4  5  5  6  7  8  8
#>  [26]  9 10 10 11 11 11 11 12 13 14 15 15 16 17 18 18 18 18 18 18 18 18 18 19 19
#>  [51] 19 19 20 20 21 21 22 22 22 23 24 24 24 24 24 24 25 26 26 26 27 27 27 28 28
#>  [76] 29 29 29 30 31 31 31 31 31 31 31 32 32 33 33 34 34 34 35 35 36 36 37 37 38
#> 
#> $Tails
#>   [1]  1  2  3  3  3  4  5  6  7  8  9 10 11 11 12 13 14 14 15 15 16 16 16 16 17
#>  [26] 17 17 18 18 19 20 21 21 21 21 21 22 22 22 22 23 24 25 26 27 28 29 30 30 31
#>  [51] 32 33 33 34 34 35 35 36 37 37 37 38 39 40 41 42 42 42 43 44 44 45 46 46 47
#>  [76] 47 48 49 49 49 50 51 52 53 54 55 55 56 56 57 57 58 59 59 60 60 61 61 62 62

^{Created on 2024-03-19 with reprex v2.1.0}

Step 2
Put the results in a nicer form, matrices. The heads and tails results are in the final named list.

# these are going to be the end result row and column names
rn <- sprintf("turn_%03d", seq.int(num_turns))
cn <- sprintf("coin_%03d", seq.int(num_coins))

# simulate the flips of all coins
# and create the results' matrices list
results_mat <- flip_all_coins(num_coins, num_turns) |> t()
results <- lapply(c("Heads", "Tails"), \(x) {
  results_mat[, x] |>
    t() |> 
    do.call(cbind, args = _) |>
    `dimnames<-`(list(rn, cn))
}) |>
  `names<-`( c("Heads", "Tails") )

results$Heads[1:5, 1:5]
#>          coin_001 coin_002 coin_003 coin_004 coin_005
#> turn_001        0        0        1        0        0
#> turn_002        0        1        2        0        0
#> turn_003        0        2        2        0        0
#> turn_004        0        2        3        0        0
#> turn_005        1        2        3        0        1
results$Tails[1:5, 1:5]
#>          coin_001 coin_002 coin_003 coin_004 coin_005
#> turn_001        1        1        0        1        1
#> turn_002        2        1        0        2        2
#> turn_003        3        1        1        3        3
#> turn_004        4        2        1        4        4
#> turn_005        4        3        2        5        4

^{Created on 2024-03-19 with reprex v2.1.0}

Answer 3

Just sample a big matrix of 1's and 0's and add them all together

First, we create a 100-by-100 matrix of flip probabilities, where each column is a coin and each row is a flip. We can calculate a simplified probability of flipping by combining the two steps of coin-selection and flipping into one fixed 'probability for flipping over'. A flip is 1, and a non-flip is 0.

set.seed(123)
# Probability of flipping over is p(flipping at all) * p(flipping over) (i.e. 0.5 * 0.4 = 0.2)
flips <- rbinom(10000, 1, 0.2)
dim(flips) <- c(100, 100)

We then add an initial row to the top of our matrix, with a random binomial values that represent the initial positions of the coins.

## Initialise flip-matrix with random coin flips
flips <- rbind(rbinom(100, 1, 0.5), flips)

To simulate the flipping, we just get the cumulative sums of each column in the matrix. In this simulation, even values are heads and odd values are tails. We can use %% to derive a logical heads/tails matrix from our numeric values:

flips <- apply(flips, 2, cumsum)

flips <- flips %% 2 == 0

Then we remove the first row (turn 0) before summing heads and tails:

## We don't tally scores for turn zero
flips <- flips[-1, ]

heads <- colSums(flips)
tails <- 100 - heads

scores <- data.frame(coin = 1:100, heads = heads, tails = tails)

head(scores)
#>   coin heads tails
#> 1    1    57    43
#> 2    2    53    47
#> 3    3    31    69
#> 4    4    60    40
#> 5    5    59    41
#> 6    6    50    50

^{Created on 2024-03-19 with reprex v2.0.2}

---

Here's the full code for convenience

set.seed(123)
# Probability of flipping over is p(flipping at all) * p(flipping over)
flips <- rbinom(10000, 1, 0.2)
dim(flips) <- c(100, 100)

## Initialise flip-matrix with random coin flips
flips <- rbind(rbinom(100, 1, 0.5), flips)

flips <- apply(flips, 2, cumsum)

flips <- flips %% 2 == 0

## We don't tally scores for turn zero
flips <- flips[-1, ]

heads <- colSums(flips)
tails <- 100 - heads

scores <- data.frame(coin = 1:100, heads = heads, tails = tails)

head(scores)
#>   coin heads tails
#> 1    1    57    43
#> 2    2    53    47
#> 3    3    31    69
#> 4    4    60    40
#> 5    5    59    41
#> 6    6    50    50

^{Created on 2024-03-19 with reprex v2.0.2}

Answer 4

The comments should be easy enough to follow.

set.seed(688396493)
N <- 100 # coins
n <- 100 # turns
ps <- 0.5 # probability of selection
pf <- 0.4 # probability of swapping sides conditioned on a flip
pt <- ps*pf # unconditional probability of swapping sides in a turn
# Initialize the cumulative heads matrix. Turns along rows, coins along columns.
heads <- matrix(0L, n, N, 0, list(paste0("turn", 1:n), paste0("coin", 1:N)))
# state after the first turn (heads = T, tails = F)
heads[1,] <- s <- sample(!0:1, N, 1)
# pre-calculate the number of coins that will swap sides on each turn
nt <- rbinom(n, N, pt)
# complete the simulation
for (i in 2:n) {
  idx <- sample(N, nt[i])
  s[idx] <- !s[idx]
  heads[i,] <- heads[i - 1,] + s
}
# calculate the tails
tails <- 1:n - heads

Results of the first 10 turns of the first 10 coins:

heads[1:10, 1:10]
#>        coin1 coin2 coin3 coin4 coin5 coin6 coin7 coin8 coin9 coin10
#> turn1      1     1     1     1     0     1     1     0     1      1
#> turn2      2     2     1     2     0     2     2     0     2      2
#> turn3      3     3     2     3     0     3     3     0     2      2
#> turn4      4     4     3     4     0     3     3     1     2      3
#> turn5      5     5     4     5     0     3     3     2     2      3
#> turn6      5     5     5     5     0     3     4     3     2      3
#> turn7      6     5     5     5     0     3     4     4     2      3
#> turn8      7     5     5     6     0     3     4     5     2      3
#> turn9      8     5     5     7     0     3     4     6     2      3
#> turn10     9     5     6     7     0     4     5     7     2      3
tails[1:10, 1:10]
#>        coin1 coin2 coin3 coin4 coin5 coin6 coin7 coin8 coin9 coin10
#> turn1      0     0     0     0     1     0     0     1     0      0
#> turn2      0     0     1     0     2     0     0     2     0      0
#> turn3      0     0     1     0     3     0     0     3     1      1
#> turn4      0     0     1     0     4     1     1     3     2      1
#> turn5      0     0     1     0     5     2     2     3     3      2
#> turn6      1     1     1     1     6     3     2     3     4      3
#> turn7      1     2     2     2     7     4     3     3     5      4
#> turn8      1     3     3     2     8     5     4     3     6      5
#> turn9      1     4     4     2     9     6     5     3     7      6
#> turn10     1     5     4     3    10     6     5     3     8      7