Since the rates of each device are independent of each other, and the user is authenticated when any one of the devices authenticates the user, we have the following information:

A user is rejected if and only if all three devices reject him (him is used throughout to represent him or her), therefore, the combined rejection rate becomes:

FR = fr1 * fr2 * fr3

So we know that introducing more biometrics systems lower the overall false rejection rate.

Since the rates of each device are independent of each other, and the user is authenticated when any one of the devices authenticates the user, we have the following information:

A user is rejected if and only if all three devices reject him (him is used throughout to represent him or her), therefore, the combined rejection rate becomes:

FR = fr1 * fr2 * fr3

So we know that introducing more biometrics systems lower the overall false rejection rate.

Now, to calculate the rate that any one of these produces a false accept:

FA = 1 - Niether 1, nor 2, nor 3 produce fa = 1 - [(1 - fa1)(1 - fa2)(1 - fa3)] = 1 - [(1 - fa1 - fa2 + fa1fa2)(1 - fa3)] = 1 - (1 - fa1 - fa2 + fa1fa2 - fa3 + fa1fa3 + fa2fa3 - fa1fa2fa3) = 1 - 1 + fa1 + fa2 + fa3 - fa1fa2 - fa2fa3 - fa3fa1 + fa1fa2fa3 = fa1 + fa2 + fa3 - fa1fa2 - fa2fa3 - fa3fa1 + fa1fa2fa3