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Let be A and B two events from the same sample set. If $\space P(A)+P(B)=1$, can one say that they are opposite events?

In my thought:

$\space P(A)+P(B)=1$

$\space P(A)=1-P(B)$

So they are opposite events. But my book says no! It says that is not necessary true.

Can you explain me, why not?

Thanks

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    $\begingroup$ If the event $A$ had $P(A)=1/2$ would you say that $A$ is opposite to itself? $\endgroup$
    – Andrea Mori
    Commented Apr 9, 2012 at 21:51

3 Answers 3

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Suppose that you roll an ordinary six-sided die. Let $A$ be the event that you roll $1,2$, or $3$, and let $B$ be the event that you roll an even number ($2,4$, or $6$). $P(A)=P(B)=\frac12$, so $P(A)+P(B)=1$; are $A$ and $B$ opposite events? (By the way, a better word is complementary events.)

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If B is the complementary event of A, then they satisfy the property:

$P(A)=1-P(B)$

But the reverse (your claim) is not true. The key thing is to realize that events are not necessarily mutually exclusive (disjoint, $P(A\cap B)=\emptyset$).

If they happen to be disjoint, then your claim would indeed be true.

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  • $\begingroup$ Both the three answers were good.But this one explained with the mutually exclusive event concept. $\endgroup$
    – user24047
    Commented Apr 10, 2012 at 17:01
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Doesn't work. Throw a die. The sample space is $\{1,2,3,4,5,6\}$.

Let $A=\{1,2,3,4\}$ and $B=\{1,2\}$.

Then $P(A)+P(B)=1$ but $A$ and $B$ are not complementary events.

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