S/N Formula with probability
The OP is looking for an S/N formula, as per the first paragraph of their statement, and also has a specific question about a nomograph figure they though might be relevant. These two things are addressed separately.

Any SNR (aka S/N) formula will necessarily depend on the specific circumstances, e.g., how is the measurement noise distributed, is it homoscedastic or not, what are the characteristics of the signal, and so on. Since none of that is given, I will do one of the simplest possible examples and leave it to the OP to consider their actual scenario.

First, assume the measurement noise is additive, white Gaussian noise with zero mean. Also assume the noise is homoscedastic, i.e., its population standard deviation, \$\sigma \$, is constant. In the absence of noise, assume the system response, denoted by \$\mu \$, is constant. For illustrative purposes, let \$\mu = 5\$ and \$\sigma = 1\$.

Let p be the probability of false positives, i.e., false alarms, and q be the probability of false negatives. Then \$1-q \$ is the detection probability. The figure below depicts the situation where \$p = 0.001 \$ and \$1-q = 0.98 \$, i.e., \$q = 0.02 \$ .



The decision level (aka decision threshold, among many other names) is the vertical blue line at \$\mu + z_p\sigma \$, where \$z_p \$ is the critical z value for p. It is easily computed via \$z_p = -normsinv(p) \$ in Excel or via lookup table. For \$p = 0.001 \$, \$z_p = 3.090 \$, so the decision level is at 8.090. The lower (black) Gaussian is the operative distribution when only noise is present and a random sample has only 0.1% probability of being above the decision level, i.e., of being a false alarm.

For \$q = 0.02 \$, \$z_q = -normsinv(q) \$, so \$z_q = 2.054 \$. Therefore, \$\mu + (z_p + z_q)\sigma \$ is where the center of the upper (red) Gaussian must be located in order for random samples from it to have 98% probability of being above the decision level, i.e., of being detected. The center of the upper Gaussian distribution is the true value of the noiseless signal and it equals 10.144 in the present example. Therefore, amplitude SNR = \$(z_p + z_q)\sigma /\sigma = (z_p + z_q) \$ and SNR = \$(z_p + z_q)^2 = 26.46 \$, i.e., 14.2 dB.

Second, the OP’s figure pertains to the performance of a quadrature receiver with incoherent detection and has nothing to do with shifted Gaussian distributions in the above figure. Whalen 1, as his Fig. 7.5 on page 205, gives the exactly equivalent figure, except with axes swapped and his figure has curves for both coherent (signal completely known) and incoherent (signal known except for phase) detection:



Whalen gives the equations for probability of false alarm, \$P_{fa}\$, and probability of detection, \$P_D\$. These are based on the Rayleigh and Rician distributions.

The Rayleigh distribution is given by
$$p_q(q) =  \frac{q}{\sigma_T^2}  e^{-q^2 / 2\sigma_T^2}  $$

where \$T\$ = sampling period, \$N_0\$ = unilateral power spectral density of white Gaussian noise, \$\sigma_T^2 =N_0 T/4\$, \$E = A^2 T/2\$, and \$SNR = E/N_0\$.

Then Whalen’s equation 7-24 is
$$ P_{fa} =  \int\limits_{\eta }^{\infty} p_q(q)  \mathrm dq = e^{-\beta^2 / 2}$$

where \$\beta = \eta / \sigma_T\$ is a normalized detection threshold. With \$z \equiv q/\sigma_T\$ and \$\alpha^2 = 2E/N_0\$, Whalen’s equation 7-27 is

$$ P_D =  \int\limits_{\beta }^{\infty} z e^{-(z^2 + \alpha^2)/2} I_0(\alpha z) \mathrm dz $$ 

where \$I_0(x)\$ is the modified Bessel function of zero order. As Whalen says, this is the Marcum Q function. Since there is no closed form for the Marcum Q function, this explains the use of a nomograph back in 1971.

Reference:


A.D. Whalen, Detection of Signals in Noise, Academic Press, New York, ©1971, pp. 202-205.