Let E and F be two independent events. The probability that both E and F happen is 1/12 and the probability that neither E nor F happens is 1/2, then a value of P(E)P(F) is.

Let P(E) = x, P(F) = y
P(E∩F) = 1/12 ⇒ P(E)P(F) = 1/12
xy = 1/12
P(E'∩F') = 1/2 ⇒ (1 - P(E))(1 - P(F)) = 1/2
⇒ (1 - x)(1 - y) = 1/2
1 - x - y + xy = 1/2
x + y = 1 + xy
x + y = 1 + 1/12 = 13/12
(x - y)² = (x + y)² - 4xy
(x - y)² = (13/12)² - 4(1/12)
(x - y)² = 169/144 - 48/144 = 121/144
(x - y)² = 121/144
x - y = ±11/12
2x = 13/12 ± 11/12
If 2x = 24/12 = 2, x = 1
If 2x = 2/12 = 1/6, x = 1/12
If x = 1, y = 1/12
If x = 1/12, y = 1
x + y = 7/12
xy = 1/12
x = 1/3, y = 1/4 or x = 1/4, y = 1/3
xy = 1/12
4/3