time, and registered the numbers of heads which appeared each time. Now the probability of obtaining 10, 9, 8, 7, &c., heads is proportional to the number of combinations of 10, 9, 8, 7, &c., things out of 10 things. Consequently the results ought to approximate to the numbers in the eleventh line of the Arithmetical Triangle. I made altogether 2048 throws, in two sets of 1024 throws each, and the numbers obtained are given in the following table:—
| Character of Throw. | Theoretical Numbers. | First Series. | Second Series. | Average. | Divergence. | |||
|---|---|---|---|---|---|---|---|---|
| 10 | Heads | 0 | Tail | 1 | 3 | 1 | 2 | + 1 |
| 9 | " | 1 | " | 10 | 12 | 23 | 17 1 / 2 | + 7 1 / 2 |
| 8 | " | 2 | " | 45 | 57 | 73 | 65 | + 20 |
| 7 | " | 3 | " | 120 | 129 | 123 | 126 | + 6 |
| 6 | " | 4 | " | 210 | 181 | 190 | 185 1 / 2 | – 25 |
| 5 | " | 5 | " | 252 | 257 | 232 | 244 1 / 2 | – 7 1 / 2 |
| 4 | " | 6 | " | 210 | 201 | 197 | 199 | – 11 |
| 3 | " | 7 | " | 120 | 111 | 119 | 115 | – 5 |
| 2 | " | 8 | " | 45 | 52 | 50 | 51 | + 6 |
| 1 | " | 9 | " | 10 | 21 | 15 | 18 | + 8 |
| 0 | " | 10 | " | 1 | 0 | 1 | 1 / 2 | – 1 / 2 |
| Totals ... ... | 1024 | 1024 | 1024 | 1024 | – 1 |
The whole number of single throws of coins amounted to 10 × 2048, or 20,480 in all, one half of which or 10,240 should theoretically give head. The total number of heads obtained was actually 10,353, or 5222 in the first series, and 5131 in the second. The coincidence with theory is pretty close, but considering the large number of throws there is some reason to suspect a tendency in favour of heads.
The special interest of this trial consists in the exhibition, in a practical form, of the results of Bernoulli’s theorem, and the law of error or divergence from the mean to be afterwards more fully considered. It illustrates the connection between combinations and permutations, which is exhibited in the Arithmetical Triangle, and which underlies many important theorems of science.
Probable Deductive Arguments .
With the aid of the theory of probabilities, we may extend the sphere of deductive argument. Hitherto we have treated propositions as certain, and on the hypothesis of certainty have deduced conclusions equally certain. But the information on which we reason in ordinary life is seldom or never certain, and almost all reasoning is really a question of probability. We ought therefore to be fully aware of the mode and degree in which deductive reasoning is affected by the theory of probability, and many persons may be surprised at the results which must be admitted. Some controversial writers appear to consider, as De Morgan remarked,7 that