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If the probability of two events are equal, can we conclude that the events are equal? In other words how does one show that the event of getting at most 3 x's is the same as the sum of the events of getting exactly 3 x's and at most 2 x's.

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What you wrote after In other words is not related to what you wrote before it. –  Did Sep 22 '11 at 15:05
    
"the sum of the events of" - Are you trying to add events? Do you mean union of these events? Or, do you mean that the probabilities are to be added? –  Srivatsan Sep 22 '11 at 15:05
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"If the probability of two events are equal, can we conclude that the events are equal?" - No. The events of a dice throw being a $1$ and a dice throw being a $2$ are not equal, even though the probabilities are (usually) the same. –  TMM Sep 22 '11 at 15:07
    
Events are sets. From equal probability you can't conclude equality of events. From equality of events you certainly can conclude equal probability. The event of at most $3$ $x$'s is the disjoint union of the events exactly $3$ and at most $2$. You get at most $3$ Aces if you get exactly $3$ or at most $2$. –  André Nicolas Sep 22 '11 at 15:08
    
"If the probability of two events are equal, can we conclude that the events are equal?" - No. To give you a really simple analogy, remember that you cannot say that two sets are equal just because their cardinalities (sizes) match (e.g., $\{1,2\} \neq \{3,4\}$). Similarly, just because two events happen to have the same probability, they are not necessarily the same. –  Srivatsan Sep 22 '11 at 15:17

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NO, same probability is not used to define "equality" of events. Instead, just say "equal probability", or be fancy and say "equiprobable".

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