The below mentioned article throws light upon the application of law of probability on coins.
Laws of Probability enable one to forecast the occurrence of any specified event. Thus if a coin is tossed 100 times it is likely that it will fall ‘head’ 50 times and ‘tail’ 50 times because the chances of a coin falling head or tail are equal i.e., 1: 1. The result will be more accurately 1: 1 the greater number of times one tosses the coin. Accuracy in Biometry always depends on the large number of individuals one considers. In order to get reliable statistics the sample size must be fairly large.
If instead of taking a single coin one takes two coins at a time, it will be found that the ratio of Two Heads. One Head the Other Tail: Two Tails will be as 1:2:1 (Fig. 823). Probability in such cases is given by the coefficient of the expansion of the binomial formula (a + b)n where n is the number of objects. If such coefficients are plotted on a graph paper, a normal curve will be obtained. Thus, if four coins are tossed at a time the probabilities will be given by the coefficients in (a + b)4, i.e., a4 + 4a3b + 6a2b2 + 4ab3 + b4.
This means that the proportion of 4 Heads: 3 Heads and 1 Tail: 2 Heads and 2 Tails: 1 Head and 3 Tails: 4 Tails will be as 1: 4: 6: 4: 1. In the beet sugar content normal curve the broken line is obtained by plotting the coefficients of the expansion of (a + b)20 and the two curves match very well indeed!
There can be an interesting application of this law in forecasting the composition of human families. The probability of any particular child of a family being a boy or a girl is 1: 1 just as in the case of head or tail of a coin. So, if we collect the statistics of families having four children each, we are likely to find the proportion of boys and girls in the same way.
It is probable that out of every 16 families having 4 children each, 1 will have all boys, 4 will have 3 boys and 1 girl, 6 will have 2 boys and 2 girls, 4 will have 1 boy and 3 girls and the last family will have all girls. The numbers are obtained from the coefficient of the expansion of (a + b)4 just as in the case of the coins.