P(G or C) = P(G) + P(C) - P(G and C)

P(G or C) = P(G) + P(C) - P(G and C)

When we look for the probability of one event or the other happening we need to add the chance of the first happening with the chance of the second happening, then we need to subtract the probability of both of them happening in order not to over-estimate the chances.

Example:
We are in a store where detailed tracking can tell us the chances of...
a person buying gum P(G) is 0.4
a person buying chocolate P(C) is 0.7
and a person buying both P(G and C) is 0.2

What is the probability that a person buys gum or chocolate?
P(G or C) = P(G) + P(C) - P(G and C)

P(G or C) = 0.4 + 0.7 - 0.2 = 0.9

There is a 90% chance of a customer buying either of those products.
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The chances that a customer does not buy gum or chocolate is the complement.

P(not G and not C) = 1 - P(G or C)

or 1 - 0.9 = 0.1 so there is a 10% chance the customer will not buy either...