Event A and B are independent of each other...
What does independent mean here? How would it change if it said dependent. The full question continues as:
The probability of event A occurring is 3x as likely as event B occurring. The probability that neither event A nor event B will occur is 7/27. What is the probability of event A occurring?

- anonymous

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- schrodinger

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- anonymous

If two events A and B are independant then
P(A intersection B)=P(A)P(B)

- anonymous

Just take P(B) as x P(A)=3x.
Apply the condition and solve for x.

- anonymous

phi the venn diagram i think is wrong, independant does not mean that it has no element in common.

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- anonymous

1-P(A union B)=7/27.
P(A union B)=20/27.
P(A)+P(B)-P(A intersection B)=20/27.
x+3x-3x^2=20/27.
Solve for x.

- anonymous

@Phi, 15/27 is not a possible answer.
Neither of you are making very much sense. I don't understand what shank means by intersection or union

- phi

yep, thanks !

- Directrix

Definition: Two events, A and B, are independent if the fact that A occurs does not affect the probability of B occurring.
Some other examples of independent events are:
Landing on heads after tossing a coin AND rolling a 5 on a single 6-sided die.
Choosing a marble from a jar AND landing on heads after tossing a coin.
Choosing a 3 from a deck of cards, replacing it, AND then choosing an ace as the second card.
Rolling a 4 on a single 6-sided die, AND then rolling a 1 on a second roll of the die.
http://www.mathgoodies.com/lessons/vol6/independent_events.html

- anonymous

Thanks directrix. Could you give some examples of dependent then?

- anonymous

how can you have a union or intersection of a probability? I thought that was in sets.

- anonymous

Please help me.

- phi

|dw:1328149905682:dw|
shankvee definitely has the right idea here

- anonymous

The picture means nothing to me because I have no idea what you are talking about in the first place. Please look at my last inquiry that asked what union and intersection of probability is.

- anonymous

Please don't give up on me. I'm getting frustrated with this problem...

- phi

maybe sat can add some insight??

- anonymous

I wish asnaseer was here. Do you know him?

- anonymous

Could you at least tell me what intersection and union are? I know what they are in sets but I don't see how it can relate to probability.

- phi

I think of it this way. You have a universe of events (that is the box) which adds up to 1
events are regions of the universe.
Here's a short write up
http://www.stat.yale.edu/Courses/1997-98/101/probint.htm

- phi

Here is a simple example
equal number of boys and girls in school
probability of selecting girl if 1/2
some students play tennis, some play chess
the students who play tennis overlap with boys and girls
what's the probability of choosing a student who plays chess and plays tennis
a venn diagram helps you visualize the problem
|dw:1328150687630:dw|

- phi

btw, prob of A occurring is 2/3

- anonymous

I still don't understand how that helps me. Back to the wording of the original problem; probability of a is 3x as likely as b occurring so I did
a = 3b
but how do i write "The probability that neither event A nor event B will occur is 7/27"
in an equation so that i can do systems?

- anonymous

your answer is correct. but i dont see how you got it, I'm not one where drawing a picture to explain your words helps much. Sure if I have a trig problem I'll draw the triangle out, but that is just to remember what the sides or angles are in relation to one another.

- anonymous

what is a disjoint event, the example it gives contradicts its definition because it says when two events have no outcomes in common the events are disjoint, but the example it gives only has one event and it says it is disjoint. To what?

- phi

"neither event A nor event B will occur is 7/27" means that A, B, or both occur 20/27 times
Prob (A) + Prob(B) - Prob( both A and B) = 3b + b - b*3b
where the x*3x comes from the statement that A and B are independent.
As pointed out above, independent means Prob(A and B) = P(A)*P(B)= 3b*b= 3b^2

- phi

we subtract Prob(both A and B) because we would be double counting if we don't

- anonymous

so does P(A) + P(B) = 20/27?

- phi

almost. if your were counting tennis players and chess players it would add up correctly. But if someone plays both chess and tennis you have to subtract the double counting.
so
Prob (A) + Prob(B) - Prob( both A and B) = 20/27

- anonymous

I see. And the probability of A and B is like saying what is the probability of this happening and then this happening. So it would be mutliplication of the two probabilities?

- phi

yes. I think events are "randomly pick" out of the universe. what's the probability of selecting something out of the A box or B box. And somethings can be in both.

- anonymous

so say 1/4 prob. of someone in chess, and 1/5 prob. someone in tennis
you would do 1/4 + 1/5 = 9/20 - 1/20 = 8/20?

- phi

if they are independent, which is a reasonable assumption.

- anonymous

what about if they were dependent, how would that change, what does that really even mean?

- phi

Dependent means the probability of event B changes (depends) on event A occurring

- anonymous

okay

- anonymous

so now we have
P(A) = P(3B)
and
P(A or B) - P(A and B) = 20/27 right?
This is the same as
P(3B or B) - P(3B and B) = P(3B) + P(B) - P(3B^2) = 20/27
Where do we go from here?

- anonymous

3B^2 - 4B + 20/27 = 0?

- anonymous

With that we get 10/9 and 2/9. 10/9 is impossible so it has to be 2/9 which is the probability of B occurring. Since A = 3B then A = 3 x 2/9 = 6/9 = 2/3!!!!!!
I got the answer!!!
Thank you thank you thank you thank you!

- phi

thanks, but I'm learning this too...

- anonymous

what do you mean?

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