I struggled a bit with understanding the mathematics in this question, but I think I found a more intuitive way to explain the solution.

Attachment:

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Let the

RED area equal

r, or

P(G), and let the

BLUE area equal

s, or

P(H).

The

PURPLE area, therefore, indicates the probability that

G and

H both occur, or

P(G intersection H).

Since we're trying to find "The probability that either

G will occur or

H will occur, but not both," then we are trying to find the

RED area and the

BLUE area, but NOT the

PURPLE area. To write this in probability terms, we would write P(

G union

H) - P (

G intersection H) =

P(G) -

P (G intersection H) +

P(H) -

P (G intersection H)The

RED area is P(G)-P(G intersection H), or r-rs. The

BLUE area is P(H)-P(G intersection H), or s-sr. You then simply add those too formulas together, resulting in r - rs + s - rs, which simplifies to r + s - 2rs.