Find P((A∪B)'): Mutually Excl. & P(A)=P(B) = 1/5

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  • Thread starter mathlearn
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In summary, if the two events A & B are mutually exclusive and $P(A) = P(B) =\frac{1}{5}$, then the probability of the complement of their union, $P((A∪B)')$, is $\frac{3}{5}$. This can be found by using the complement rule and the fact that mutually exclusive events have a combined probability of their individual probabilities.
  • #1
mathlearn
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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 :)
 
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  • #2
$(A\cup B)'= A' \cap B'$
 
  • #3
HallsofIvy said:
$(A\cup B)'= A' \cap B'$

$P(A') = P(B') =\frac{4}{5}$

With that known ,

$ A' \cap B' = \frac{1}{5}$

Correct? :)
 
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  • #4
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 :)

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 $A'\cap B'$.
 
  • #5
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 $A'\cap B'$.

$P(A\cup B)=P(A)+P(B)$
$P(A\cup B)=P(\frac{1}{5})+P(\frac{1}{5})$
$P(A\cup B)=P(\frac{2}{5})$

$P((A∪B)')$ This implies that everything except the addition of the probability of the two events $P(\frac{3}{5})$, Correct ?
 
  • #6
mathlearn said:
$P(A\cup B)=P(A)+P(B)$
$P(A\cup B)=P(\frac{1}{5})+P(\frac{1}{5})$
$P(A\cup B)=P(\frac{2}{5})$

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)

$P((A∪B)')$ This implies that everything except the addition of the probability of the two events $P(\frac{3}{5})$, Correct ?

Yep. (Nod)
More specifically:

$P((A∪B)') = 1 - P(A∪B) = 1 - \frac 25 = \frac 35$
 
  • #7
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:

$P((A∪B)') = 1 - P(A∪B) = 1 - \frac 25 = \frac 35$

(Star) What a fantastic explanation , Thank you very much (Nerd)
 

FAQ: Find P((A∪B)'): Mutually Excl. & P(A)=P(B) = 1/5

What is the probability of the complement of the union of A and B being mutually exclusive if the probabilities of A and B are both 1/5?

The probability of the complement of the union of A and B being mutually exclusive is 0. This is because the complement of a union means all the elements that are not in the union, and since A and B have the same probability and are mutually exclusive, they have no shared elements. Therefore, their complement will also have no shared elements and will be mutually exclusive.

How do you find the probability of the complement of the union of A and B?

To find the probability of the complement of the union of A and B, you can use the formula P((A∪B)') = 1 - P(A∪B). This is because the probability of the complement is equal to 1 minus the probability of the event itself. In this case, since A and B are mutually exclusive, P(A∪B) will be equal to the sum of their individual probabilities, which is 1/5 + 1/5 = 2/5. Therefore, P((A∪B)') = 1 - 2/5 = 3/5.

Can the probabilities of A and B being 1/5 change and still have the complement of their union be mutually exclusive?

No, the probabilities of A and B must remain the same (1/5) in order for the complement of their union to be mutually exclusive. If the probabilities were to change, then the events would no longer be mutually exclusive and their complement would not be mutually exclusive either.

Can two events with the same probability always be considered mutually exclusive?

No, two events with the same probability cannot always be considered mutually exclusive. In order for events to be mutually exclusive, they must have no shared elements. This means that even if two events have the same probability, if they have some shared elements, they cannot be considered mutually exclusive. In this case, since A and B have the same probability and are mutually exclusive, they must have no shared elements.

What is the relationship between the probability of A and B being equal and their complement being mutually exclusive?

The relationship between the probabilities of A and B being equal and their complement being mutually exclusive is that if the probabilities of A and B are equal, then their complement will be mutually exclusive. This is because when the probabilities are equal, the two events have the same chance of occurring and therefore have no shared elements. This means that their complement will also have no shared elements and will be mutually exclusive.

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