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I am new to the probability area as well but am trying to study it. I am not sure when you can use multiplication and skip inclusion-exclusion.
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You can just multiply and skip inclusion-exclusion only when the events are
mutually exclusive, meaning that only 1 player can have the hand. An example would be the probability of an opponent holding AA when you also hold AA. Since only one opponent can have AA with you, you can just multiply the number of opponents by the probability that just one opponent has it, N*6/1225. Note that we are actually
adding the N probabilities in this case to compute the probability of a union. If more than one opponent can have the hand, then this would double count the times that 2 players have the hand, triple count the times 3 players have it, etc.
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Why couldn't you say the chance of no one having a dominating hand is
c(50,2)-24/c(50,2) and then raise this to the ninth power for eight other people.
This gives you the percentage no one dominates, subtract 1 and you have the percentage of hands that at least one person dominates.
This comes out to .16312 percent, I am confused at why there is a difference.
Thank you,
Cobra
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Now this method is not exact because the player's hands are not
independent. That is, the event of one player having or not having a dominating hand changes the probability of the other players having a dominating hand, so you cannot simply multiply these probabilities together. You must instead multiply the conditional probabilities that each player does not have the hand given that other players do not have the hand. Your (1225-24)/1225 only applies to the first player, while the remaining conditional probabilities will be slightly different.
Note that these are two separate issues. You can read more about this important topic in
this recent post. Please follow the links in that post also.