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For disjoint events A and B, P(A or B)=P(A)+P(B). This is called the addition rule for disjoint events and it generalizes to more events as long as they are all disjoint.

But something is bothering me. What if A is getting heads on a coin flip, and B is getting a 2 on a die roll, C is getting a heart from drawing a card, etc. Are these not disjoint events?

I'm worried that their probabilities will soon sum to be larger than 1 (once I add enough disjoint events).

I'm sure I'm misunderstanding something here.

EDIT: As a simplified statement, imagine I have a penny, a dime, and a quarter, and I want to toss all 3 and determine the probability of P(H or H or H). I'm worried about this calculation that says P(H)+P(H)+P(H)=1.5. Obviously the probability of getting a heads on the penny or a heads on the dime or a heads on the quarter is not 1.5. What am I misunderstanding about the formula or about disjoint events?

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    $\begingroup$ Since it is possible to get H AND H AND H, the events aren't disjoint! $\endgroup$
    – jbowman
    Commented Jan 11, 2023 at 21:34
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    $\begingroup$ "Getting heads" is the event "the coin flip is heads and the die and card could be anything." Getting a 2 on the die is the event "the die shows a 2 and the coin and card could be anything." These events are not disjoint: in common they have "the coin is heads, the die is 2, and the card could be anything," a set which is nonempty. $\endgroup$
    – whuber ♦
    Commented Jan 11, 2023 at 22:03
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    $\begingroup$ Those three examples are presumably independent events. They are not disjoint as there is a positive probability of all three happening of $\frac12\times \frac16\times \frac14=\frac1{48}$ if they are fair. See stats.stackexchange.com/questions/380789/… for somebody confused about independent $\endgroup$
    – Henry
    Commented Jan 12, 2023 at 1:55

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