# when do we say two events are equal

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
"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