Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question
1  
If the event $A$ had $P(A)=1/2$ would you say that $A$ is opposite to itself? –  Andrea Mori Apr 9 '12 at 21:51
add comment

3 Answers 3

up vote 4 down vote accepted

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.

share|improve this answer
    
Both the three answers were good.But this one explained with the mutually exclusive event concept. –  João Apr 10 '12 at 17:01
add comment

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.)

share|improve this answer
add comment

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.