The discussion revolves around calculating the probability of the complement of the union of two mutually exclusive events, A and B, given that both events have a probability of 1/5. Participants explore the application of probability rules and seek clarification on the steps involved in the calculation.
Participants generally agree on the calculation of $P(A \cup B)$ and the application of the complement rule, but there is some uncertainty regarding the interpretation and use of $A' \cap B'$ in the context of the problem.
Some participants express confusion about the relationship between the complements and the union of the events, indicating potential gaps in understanding the application of probability rules.
HallsofIvy said:$(A\cup B)'= A' \cap B'$
mathlearn said:If the two events A & B are mutually exclusive and $P(A) = P(B) =\frac{1}{5}$, then find $P((A∪B)')$.
I cannot recall anything on this, help would be appreciated :)
I like Serena said:Mutually exclusive means $P(A\cup B)=P(A)+P(B)$.
We can combine it with the complement rule that states that $P(E')=1-P(E)$. (Thinking)
I'm afraid I don't see what we can do with $A'\cap B'$.
mathlearn said:$P(A\cup B)=P(A)+P(B)$
$P(A\cup B)=P(\frac{1}{5})+P(\frac{1}{5})$
$P(A\cup B)=P(\frac{2}{5})$
$P((A∪B)')$ This implies that everything except the addition of the probability of the two events $P(\frac{3}{5})$, Correct ?
I like Serena said:That should be:
$P(A\cup B)=P(A)+P(B)$
$P(A\cup B)=\frac{1}{5}+\frac{1}{5}$
$P(A\cup B)=\frac{2}{5}$
since $P$ is a function of an event that yields a number. (Nerd)
Yep. (Nod)
More specifically:
$P((A∪B)') = 1 - P(A∪B) = 1 - \frac 25 = \frac 35$