Mutually exclusive events and the addition rule
Mutually exclusive events can be added. Overlapping events cannot, unless you subtract the double count. A die shows both cases.
Two events are mutually exclusive when they cannot happen together. A die cannot show 1 and 2 on the same roll. If you want the chance of one or the other, and they cannot overlap, you add. If they can overlap, adding counts the overlap twice. Most everyday "or" sentences are the second kind, and people add anyway.
This is for adults 18 and older. The die is the whole lesson. It is not a system. Laws on wagering differ by place.
Adding when the overlap is empty
A fair die has six faces, each with probability 1/6. The chance of a 1 is 1/6. The chance of a 2 is 1/6. A 1 and a 2 cannot arrive together, so the chance of a 1 or a 2 is 1/6 plus 1/6, which is 1/3. Three faces, 1 or 2 or 3, give 1/2. Six faces give 1. The additions stay honest because each face was counted once.
This is the addition rule in its clean form. It is the mirror of the multiplication you use for independent events that must happen together. "Or" with no overlap adds. "And" for independent steps multiplies. Mixing those two words is how a plausible fraction goes wrong.
Adding when the overlap is real
Ask for an even number or a number greater than 4. Even faces are 2, 4 and 6, so the chance is 1/2. Faces greater than 4 are 5 and 6, so the chance is 2/6, which is 1/3. Add those and you get 1/2 plus 1/3, which is 5/6. The true list is 2, 4, 5 and 6. That is 4 faces out of 6, which is 2/3. The extra 1/6 in the sloppy sum is the face 6, counted in both groups.
The repair is to subtract the overlap. Chance of even, plus chance of greater than 4, minus chance of both, which is the chance of 6, 1/6. So 1/2 plus 1/3 minus 1/6 equals 3/6 plus 2/6 minus 1/6, which is 4/6. That matches the list. If you cannot name the overlap, you are not ready to add.
Where bets hide the overlap
A roulette chip on red and a chip on even are not mutually exclusive. Red even numbers pay both. Red odd numbers pay one. Black even numbers pay one. The layout does not warn you that "covering more" double-counts some pockets and misses the zeros entirely. Inside and outside bets are a map of coverage. Coverage is not a sum of the printed chances unless you have removed the overlaps and the gaps.
Sports markets do this too. A team win and an over are not exclusive, and they are not independent either. Adding their implied chances, or multiplying them without a reason, produces a number with no job. Parlays fail in the multiplication direction. Sloppy "or" bets fail in the addition direction. Both feel like arithmetic.
A check before you trust a total
List the outcomes in the first event. List the outcomes in the second. Highlight anything in both lists. If the highlight is empty, add. If it is not, add and then subtract the highlight, or just count the combined list once. Combinations are the same discipline with a deck: decide what you are counting before you divide.
A clean sum is still not a price. The chance of rolling a 1 or a 2 is 1/3. A game that pays less than 2 to 1 on that event is a worse-than-fair offer, even though your addition was perfect. Correct odds you do not get paid for are a description, not a wage. If the describing has become a reason to stake bill money, stop. Local help beats a prettier fraction.
Frequently asked questions
When can I add two chances? When the events cannot happen together, or when you subtract the overlap if they can. Adding two overlapping chances counts the overlap twice.
Is "1 or 2" on a die one third? Yes, for a fair six-sided die. Each face is 1/6, they cannot overlap, and 1/6 plus 1/6 is 1/3.
Does a correct sum mean the bet is fair? No. The sum is the chance. The payout is the price. A house edge is a price that sits below the fair return for that chance.