I do understand what mutually exclusive events mean (i.e. $ A \cap B = \emptyset$). But I came across this example in this book

The following table presents probabilities for the number of times that a certain computer system will crash in the course of a week. Let A be the event that there are more than two crashes during the week, and let B be the event that the system crashes at least once. Find a sample space. Then find the subsets of the sample space that correspond to the events A and B. Then find P(A) and P(B).

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and the author states that

The events “3 crashes happen” and “4 crashes happen” are mutually exclusive.

Why these events are mutually exclusive? If a "4 crashes happen" event happens means (to me) the "3 crashes happen" event happens for sure. Am I right?

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    $\begingroup$ The events: exactly $3$ crashes happen' and exactly $4$ crashes happen' are disjoint. That is what they mean here. $\endgroup$ – drhab Aug 25 '14 at 14:59
  • $\begingroup$ But there is no exactly in the example. Also, why they are disjoint, this is my question actually. $\endgroup$ – CroCo Aug 25 '14 at 15:01
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    $\begingroup$ @CroCo: it is implied by the meaning of the value itself. For example, if I ask you how many times you brushed your teeth today, if you say "once," then that doesn't mean you brushed your teeth both zero times and one time. If you say "twice," it doesn't mean you brushed your teeth once and twice simultaneously. We do not need to use the word "exactly." $\endgroup$ – heropup Aug 25 '14 at 15:04
  • $\begingroup$ @heropup, but we here talk about how many times an even occurs. The output of this experiment is the number of crashes, so there is nothing in this example states that the events in the sample space are mutually exclusive. $\endgroup$ – CroCo Aug 25 '14 at 15:10
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    $\begingroup$ The events are mutually exclusive because, as stated by @heropup, of the concept of conceiving both events. It is not a conclusion of the sample space: both events are always mutually exclusive, and the context or experiment or sample space or anything does not matter... the statement holds. $\endgroup$ – cjferes Aug 25 '14 at 15:45

Clearly the number recorded is the count of crashes in a week.   The tally or total number of crashes that happen.

If the count of crashes that happen in a week is $4$, then the count of crashes that happen in that week is not $3$.   They are thus mutually exclusive events.

Otherwise we would measure $100\%$ for "at least $0$", instead we have $60\%$ so it's obviously for "exactly $0$".


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