The mean value of the spin is given by averaging 1 and −1 with the weights p and 1 − p, so the mean value is 2 p − 1. But this average is the same for all spins, and is therefore equal to H. H = 2 p − − 1 = 1 − − e − − 2 β β J H 1 + e − − 2 β β J H = tanh ⁡ ⁡ ( β β J H )